Diagrammatic Proofs: An Experiment in Notation
Traditional
Proposition. Let % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \ra and % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \rb be relations over some set A % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } A so that % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \rb is transitive and reflexive. Also assume the lifting condition that if bcd % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } b \rb c \ra d then exists c % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } c' so that bcd % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } b \ra c' \rb d. Let (,) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \fa{\ra,\rb} be the relation over A % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } A which includes a tuple (a,b) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } (a, b) iff exists a sequence {xn} % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \{x_n\} in A % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } A with x1=a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x_1 = a and xn=b % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x_n = b and for each n % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } n either xnxn+1 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x_n \ra x_{n+1} or xnxn+1 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x_n \rb x_{n+1}. Let , % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \fb{\ra,\rb} be the relation over A % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } A which includes a tuple (a,b) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } (a, b) iff exists a sequence {xn} % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \{x_n\} in A % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } A with x1=a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x_1 = a and where for each n % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } n we have that xnxn+1 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x_n \ra x_{n+1}, and finally where xnb % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x_n \rb b. Then (,)=, % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \fa{\ra,\rb} = \fb{\ra,\rb}. Proof.
(⇐)
Trivial
(⇒)
Take a(,)b % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } a \fa{\ra,\rb} b and witness sequence {xn}1nN % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \{x_n\}_{1 \leq n \leq N}. Since % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \rb is transitive we may WLOG assume that there are no % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \rb-chains in {xn} % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \{x_n\} of length greater than one. Now proceed by induction on N % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } N: Base case N=1 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } N = 1. Then a=b % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } a = b and so a,b % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } a \fb{\ra,\rb} b holds by reflexivity of % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \rb. Inductive case. Then we know x1(,)xN1 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x_1 \fa{\ra,\rb} x_{N{-1}} and so by the inductive hypothesis have that x1,xN1 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x_1 \fb{\ra,\rb} x_{N{-1}}. Hence exists a sequence {yk} % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \{y_k\} where y1=x1=a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } y_1 = x_1 = a and consisting of a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \ra-chain terminated by ykxN1 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } y_k \rb x_{N{-1}}. Now proceed by cases. If xN1xN % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x_{N{-1}} \rb x_N then by transitivity of % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \rb we have ykxN % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } y_k \rb x_N and we are done. If on the other hand xN1xN % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x_{N{-1}} \ra x_N then we have ykxN1xN % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } y_k \rb x_{N{-1}} \ra x_N and by applying the lifting condition we know that exists some x~ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \tilde x so that ykx~xN % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } y_k \ra \tilde x \rb x_N, and we are again done.
Diagrammatic
Proposition. Let % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \ra and % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \rb be relations over some set A % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } A so that % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \rb is transitive and reflexive. Also assume the lifing condition that for all x,y,zA % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x, y, z \in A, and where % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \bull indicates an element of A % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } A that we are neglecting to give a name to. (Ie, presence of % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \bull is an existential statement) Now let x(,)y % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x \fa{\ra,\rb} y iff xy % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x \sim \bull \sim \bull \sim \cdots \sim \bull \sim y where each % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \sim stands for either % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \ra or % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \rb. Empty % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \sim-sequences are permitted. Also let x,y % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x \fb{\ra,\rb} y iff xy % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x \ra \bull \ra \bull \ra \cdots \ra \bull \rb y Empty % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \ra-sequences are permitted; the smallest witness to x,y % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x \fb{\ra,\rb} y is xy % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x \rb y. Then as relations we have (,)=, % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } {\fa{\ra,\rb}} = {\fb{\ra,\rb}} Proof.
(⇐)
Trivial
(⇒)
Take x(,)y % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x \fa{\ra,\rb} y. Then exists a path of elements of A % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } A along the grid If we choose a path of size zero, then x=y % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x = y and so by reflexivity of % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \rb we get the chain xy % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x \rb y completing the proof. To address paths of nonzero size, we will proceed by choosing a specific example path on which to work; the choice path will be generic enough to easily extend to a proof covering all cases. We choose this path: now, in left-to-right order, we repeatedly apply transitivity of % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } \rb to obtain the following cyan-colored arrows, and repeatedly apply the lifting condition to obtain the following red-colored arrows: In the end what we produce is a chain xy % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } % category names \newcommand{\cat}[1]{{ \sf{#1} }} % more shorthands \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \newcommand{\ra}{ \rightsquigarrow } \newcommand{\rb}{ \Rightarrow } \newcommand{\fa}[1]{ \mathop{({#1})} } \newcommand{\fb}[1]{ \mathop{\langle {#1} \rangle} } x \ra \bullet \ra \bullet \ra \cdots \ra \bullet \rb y just as we desired