<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font>;<br/>



<b>set </b><font color="Maroon" title="c3">F1</font> = <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>;<br/>
<b>set </b><font color="Maroon" title="c4">F2</font> = <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>;<br/>
<a NAME="E1:4"/><i><font color="Green" title="E4">A1</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font> holds  <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Olive" title="b1">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Olive" title="b1">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Olive" title="b1">a</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>;<br/>

<a NAME="E1:4_1"/><i><font color="Green" title="E5">A2</font></i>: 
 <a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K11" title="ENS_1:func.11">fId</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="ens_1.html#D14" title="ENS_1:def.14">ENS_1:def 14</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">m</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c5">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="ref" href="ens_1.html#T48" title="ENS_1:th.48">ENS_1:48</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">m'</font> = <font color="Maroon" title="c6">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="ref" href="ens_1.html#T48" title="ENS_1:th.48">ENS_1:48</a></i>;<br/>
<i><font color="Green" title="E6">A3</font></i>:  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">m</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_1"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="cat_1.html#D2" title="CAT_1:def.2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c7">m'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D10" title="ENS_1:def.10">ENS_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D4" title="ENS_1:def.4">ENS_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <span class="p3">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="ref" href="ens_1.html#T60" title="ENS_1:th.60">ENS_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="ref" href="cat_1.html#D20" title="CAT_1:def.20">CAT_1:def 20</a></i>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="ref" href="ens_1.html#D26" title="ENS_1:def.26">ENS_1:def 26</a></i>
;<br/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c6">m</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_1"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="cat_1.html#D3" title="CAT_1:def.3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c7">m'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D11" title="ENS_1:def.11">ENS_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D5" title="ENS_1:def.5">ENS_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <span class="p3">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="ref" href="ens_1.html#T60" title="ENS_1:th.60">ENS_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="ref" href="cat_1.html#D20" title="CAT_1:def.20">CAT_1:def 20</a></i>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="ref" href="ens_1.html#D26" title="ENS_1:def.26">ENS_1:def 26</a></i>
;<br/>
<b>hence </b><a NAME="E6:4_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c5">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c5">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E4:4_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:4"/><b>then </b>
for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>  holds  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E3:4"/><b>then </b><i><font color="Green" title="E5">A4</font></i>: 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">B</font> =  <a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">A</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,$1</span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,$1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,$1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E4:4"/><i><font color="Green" title="E6">A5</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>;<br/>

<a NAME="E1:4_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c6">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c6">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:4"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:4_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">t</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">A</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span><b> such that </b><br/><a NAME="E6:4"/><i><font color="Green" title="E7">A6</font></i>: 
for <font color="Olive" title="b1">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Maroon" title="c6">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">o</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S8" title="FUNCT_2:sch.8">FUNCT_2:sch 8</a>(<a class="txt" href="yoneda_1.html#E4:4"><i><font color="Green" title="E6">A5</font></i></a>);<br/></i>
<a NAME="E7:4"/>
for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Maroon" title="c6">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Olive" title="b1">a</font>,<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>;<br/>

<a NAME="E1:4_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c7">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c7">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:4"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/>
<a NAME="E2:4_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c7">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c7">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">a</font>,<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">a</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E3:4_3"/>
<font color="Maroon" title="c6">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">a</font>,<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E6:4"><i><font color="Green" title="E7">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">t</font> = <font color="Maroon" title="c6">t</font> as    <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <b>by </b><i><a class="txt" href="yoneda_1.html#E3:4"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="nattra_1.html#D3" title="NATTRA_1:def.3">NATTRA_1:def 3</a></i>;<br/>
<a NAME="E9:4"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>  st  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>for <font color="Olive" title="b3">g</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive" title="b3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive" title="b3">g</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E1:4_4"/><i><font color="Green" title="E8">A7</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c10">g</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>;<br/>

<a NAME="E2:4_4"/><i><font color="Green" title="E9">A8</font></i>: 
 <a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K11" title="ENS_1:func.11">fId</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="ens_1.html#D14" title="ENS_1:def.14">ENS_1:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">f1</font> = <font color="Maroon" title="c7">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c9">b</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">f2</font> = <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c10">g</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<i><font color="Green" title="E10">A9</font></i>: <font color="Maroon" title="c7">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c9">b</font> = 
<font color="Maroon" title="c7">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E3:4"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c9">b</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c9">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E6:4"><i><font color="Green" title="E7">A6</font></i></a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">f1'</font> = <font color="Maroon" title="c11">f1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="ref" href="ens_1.html#T48" title="ENS_1:th.48">ENS_1:48</a></i>;<br/>
<a NAME="E7:4_4"/><i><font color="Green" title="E11">A10</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c11">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c9">b</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c11">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c9">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_2"><b>proof </b></a><div class="add">

<i><font color="Green" title="E12">A11</font></i>:  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c11">f1</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">f1</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:4_4"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="cat_1.html#D2" title="CAT_1:def.2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c13">f1'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D10" title="ENS_1:def.10">ENS_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f1</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D4" title="ENS_1:def.4">ENS_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c9">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c9">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:4_4"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c9">b</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c11">f1</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">f1</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:4_4"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="cat_1.html#D3" title="CAT_1:def.3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c13">f1'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D11" title="ENS_1:def.11">ENS_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f1</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D5" title="ENS_1:def.5">ENS_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c9">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c9">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:4_4"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c9">b</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
<b>hence </b><a NAME="E3:4_4_2"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c11">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c9">b</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c11">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c9">b</font> )
 <b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4_2"><i><font color="Green" title="E12">A11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green" title="E12">A12</font></i>: <font color="Maroon" title="c12">f2</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K20" title="ENS_1:func.20">hom?-</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#D22" title="ENS_1:def.22">ENS_1:def 22</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">f2'</font> = <font color="Maroon" title="c12">f2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="ref" href="ens_1.html#T47" title="ENS_1:th.47">ENS_1:47</a></i>;<br/>
<a NAME="E10:4_4"/><i><font color="Green" title="E13">A13</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c12">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c12">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_4"><b>proof </b></a><div class="add">

<i><font color="Green" title="E14">A14</font></i>:  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c12">f2</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">f2</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:4_4"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="cat_1.html#D2" title="CAT_1:def.2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c14">f2'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D10" title="ENS_1:def.10">ENS_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">f2</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D4" title="ENS_1:def.4">ENS_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E8:4_4"><i><font color="Green" title="E12">A12</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c12">f2</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">f2</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:4_4"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="cat_1.html#D3" title="CAT_1:def.3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c14">f2'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D11" title="ENS_1:def.11">ENS_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">f2</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D5" title="ENS_1:def.5">ENS_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E8:4_4"><i><font color="Green" title="E12">A12</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
<b>hence </b><a NAME="E3:4_4_4"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c12">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c12">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span> )
 <b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4_4"><i><font color="Green" title="E14">A14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:4_4"/><b>then </b><i><font color="Green" title="E14">A15</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c12">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c11">f1</font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4"><i><font color="Green" title="E8">A7</font></i></a>, <a class="txt" href="yoneda_1.html#E7:4_4"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">f3</font> = <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c10">g</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">f4</font> = <font color="Maroon" title="c7">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c8">a</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<i><font color="Green" title="E15">A16</font></i>: <font color="Maroon" title="c15">f3</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K20" title="ENS_1:func.20">hom?-</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#D22" title="ENS_1:def.22">ENS_1:def 22</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">f3'</font> = <font color="Maroon" title="c15">f3</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="ref" href="ens_1.html#T47" title="ENS_1:th.47">ENS_1:47</a></i>;<br/>
<a NAME="E16:4_4"/><i><font color="Green" title="E16">A17</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c15">f3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c15">f3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_6"><b>proof </b></a><div class="add">

<i><font color="Green" title="E17">A18</font></i>:  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c15">f3</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">f3</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:4_4"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="cat_1.html#D2" title="CAT_1:def.2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c17">f3'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D10" title="ENS_1:def.10">ENS_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">f3</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D4" title="ENS_1:def.4">ENS_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E14:4_4"><i><font color="Green" title="E15">A16</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c15">f3</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">f3</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:4_4"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="cat_1.html#D3" title="CAT_1:def.3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c17">f3'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D11" title="ENS_1:def.11">ENS_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">f3</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D5" title="ENS_1:def.5">ENS_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E14:4_4"><i><font color="Green" title="E15">A16</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
<b>hence </b><a NAME="E3:4_4_6"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c15">f3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c15">f3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span> )
 <b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4_6"><i><font color="Green" title="E17">A18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green" title="E17">A19</font></i>: <font color="Maroon" title="c7">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c8">a</font> = 
<font color="Maroon" title="c7">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E3:4"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E6:4"><i><font color="Green" title="E7">A6</font></i></a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">f4'</font> = <font color="Maroon" title="c16">f4</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="ref" href="ens_1.html#T48" title="ENS_1:th.48">ENS_1:48</a></i>;<br/>
<a NAME="E19:4_4"/><i><font color="Green" title="E18">A20</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c16">f4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">a</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c16">f4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_8"><b>proof </b></a><div class="add">

<i><font color="Green" title="E19">A21</font></i>:  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c16">f4</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">f4</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:4_4"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="cat_1.html#D2" title="CAT_1:def.2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c18">f4'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D10" title="ENS_1:def.10">ENS_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">f4</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D4" title="ENS_1:def.4">ENS_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E17:4_4"><i><font color="Green" title="E17">A19</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">a</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c16">f4</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">f4</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:4_4"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="cat_1.html#D3" title="CAT_1:def.3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c18">f4'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D11" title="ENS_1:def.11">ENS_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">f4</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D5" title="ENS_1:def.5">ENS_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E17:4_4"><i><font color="Green" title="E17">A19</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">a</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
<b>hence </b><a NAME="E3:4_4_8"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c16">f4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">a</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c16">f4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">a</font> )
 <b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4_8"><i><font color="Green" title="E19">A21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:4_4"/><b>then </b><i><font color="Green" title="E19">A22</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c16">f4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c15">f3</font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4"><i><font color="Green" title="E8">A7</font></i></a>, <a class="txt" href="yoneda_1.html#E16:4_4"><i><font color="Green" title="E16">A17</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>;<br/>
<a NAME="E21:4_4"/><i><font color="Green" title="E20">A23</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">b</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_9"><b>proof </b></a><div class="add">

<a NAME="E1:4_4_9"/><i><font color="Green" title="E21">A24</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_4_9_1"/>
(  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c9">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_9_1_1"><b>suppose </b></a><a NAME="E1:4_4_9_1_1"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:4_4_9_1_1"/><b>then </b><i><font color="Green" title="E22">A25</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="ens_1.html#T42" title="ENS_1:th.42">ENS_1:42</a></i>;<br/><a NAME="E3:4_4_9_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><b>hence </b><a NAME="E4:4_4_9_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4_9"><i><font color="Green" title="E21">A24</font></i></a>, <a class="txt" href="yoneda_1.html#E2:4_4_9_1_1"><i><font color="Green" title="E22">A25</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_9_1_2"><b>suppose </b></a><a NAME="E1:4_4_9_1_2"/><i><font color="Green" title="E22">A26</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c9">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>let </b><font color="Maroon" title="c19">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/><b>assume </b><a NAME="E2:4_4_9_1_2"/><i><font color="Green" title="E23">A27</font></i>: 
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span>
 ;<br/><a NAME="E3:4_4_9_1_2"/><i><font color="Green" title="E24">A28</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>;<br/><a NAME="E4:4_4_9_1_2"/><i><font color="Green" title="E25">A29</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><a NAME="E5:4_4_9_1_2"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4_9_1_2"><i><font color="Green" title="E22">A26</font></i></a>, <a class="txt" href="yoneda_1.html#E3:4_4_9_1_2"><i><font color="Green" title="E24">A28</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E6:4_4_9_1_2"/>
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E2:4_4_9_1_2"><i><font color="Green" title="E23">A27</font></i></a>, <a class="txt" href="yoneda_1.html#E4:4_4_9_1_2"><i><font color="Green" title="E25">A29</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E22:4_4"/><i><font color="Green" title="E21">A30</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_10"><b>proof </b></a><div class="add">

<a NAME="E1:4_4_10"/><i><font color="Green" title="E22">A31</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_4_10_1"/>
(  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_10_1_1"><b>suppose </b></a><a NAME="E1:4_4_10_1_1"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:4_4_10_1_1"/><b>then </b><i><font color="Green" title="E23">A32</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="ens_1.html#T42" title="ENS_1:th.42">ENS_1:42</a></i>;<br/><a NAME="E3:4_4_10_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><b>hence </b><a NAME="E4:4_4_10_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4_10"><i><font color="Green" title="E22">A31</font></i></a>, <a class="txt" href="yoneda_1.html#E2:4_4_10_1_1"><i><font color="Green" title="E23">A32</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_10_1_2"><b>suppose </b></a><a NAME="E1:4_4_10_1_2"/><i><font color="Green" title="E23">A33</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>let </b><font color="Maroon" title="c19">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/><b>assume </b><a NAME="E2:4_4_10_1_2"/><i><font color="Green" title="E24">A34</font></i>: 
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span>
 ;<br/><a NAME="E3:4_4_10_1_2"/><i><font color="Green" title="E25">A35</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><a NAME="E4:4_4_10_1_2"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4_10"><i><font color="Green" title="E22">A31</font></i></a>, <a class="txt" href="yoneda_1.html#E1:4_4_10_1_2"><i><font color="Green" title="E23">A33</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E5:4_4_10_1_2"/>
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E2:4_4_10_1_2"><i><font color="Green" title="E24">A34</font></i></a>, <a class="txt" href="yoneda_1.html#E3:4_4_10_1_2"><i><font color="Green" title="E25">A35</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E23:4_4"/><i><font color="Green" title="E22">A36</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> )
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>;<br/>
<a NAME="E24:4_4"/><i><font color="Green" title="E23">A37</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_11"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_4_11_1"/>
(  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_11_1_1"><b>suppose </b></a><a NAME="E1:4_4_11_1_1"/><i><font color="Green" title="E24">A38</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"> <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span></span>)</span> = 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E21:4_4"><i><font color="Green" title="E20">A23</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4_11_1_1"><i><font color="Green" title="E24">A38</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E23:4_4"><i><font color="Green" title="E22">A36</font></i></a>, <a class="txt" href="yoneda_1.html#E1:4_4_11_1_1"><i><font color="Green" title="E24">A38</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E22:4_4"><i><font color="Green" title="E21">A30</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>
;<br/><b>hence </b><a NAME="E3:4_4_11_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_11_1_2"><b>suppose </b></a><a NAME="E1:4_4_11_1_2"/><i><font color="Green" title="E24">A39</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:4_4_11_1_2"/><b>then </b><i><font color="Green" title="E25">A40</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="ens_1.html#T42" title="ENS_1:th.42">ENS_1:42</a></i>;<br/><a NAME="E3:4_4_11_1_2"/><i><font color="Green" title="E26">A41</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="yoneda_1.html#E23:4_4"><i><font color="Green" title="E22">A36</font></i></a>, <a class="txt" href="yoneda_1.html#E1:4_4_11_1_2"><i><font color="Green" title="E24">A39</font></i></a>, <a class="ref" href="ens_1.html#T42" title="ENS_1:th.42">ENS_1:42</a></i>;<br/> <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span></span>)</span> = 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E21:4_4"><i><font color="Green" title="E20">A23</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:4_4_11_1_2"><i><font color="Green" title="E25">A40</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">a</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E3:4_4_11_1_2"><i><font color="Green" title="E26">A41</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E22:4_4"><i><font color="Green" title="E21">A30</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>
;<br/><b>hence </b><a NAME="E5:4_4_11_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E25:4_4"/><i><font color="Green" title="E24">A42</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:4_4_12"/><i><font color="Green" title="E25">A43</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_12_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_4_12_1"/>
(  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_12_1_1"><b>suppose </b></a><a NAME="E1:4_4_12_1_1"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:4_4_12_1_1"/><b>then </b><i><font color="Green" title="E26">A44</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="ens_1.html#T42" title="ENS_1:th.42">ENS_1:42</a></i>;<br/><a NAME="E3:4_4_12_1_1"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4_12"><i><font color="Green" title="E25">A43</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E4:4_4_12_1_1"/><b>then </b><i><font color="Green" title="E27">A45</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E2:4_4_12_1_1"><i><font color="Green" title="E26">A44</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c20">x</font> = <font color="Maroon" title="c19">x</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E6:4_4_12_1_1"/><i><font color="Green" title="E28">A46</font></i>: 
<font color="Maroon" title="c10">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c9">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_12_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:4_4_12_1_1_1"/><i><font color="Green" title="E29">A47</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c20">x</font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E4:4_4_12_1_1"><i><font color="Green" title="E27">A45</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/>
<b>then </b><i><font color="Green" title="E30">A48</font></i>:  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c20">x</font></span>)</span> = 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c20">x</font>
<b>by </b><i><a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E4:4_4_12_1_1"><i><font color="Green" title="E27">A45</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>
;<br/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c20">x</font></span>)</span> = 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4_12_1_1_1"><i><font color="Green" title="E29">A47</font></i></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>
<br/>.= 
<font color="Maroon" title="c9">b</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>
;<br/>
<b>hence </b><a NAME="E4:4_4_12_1_1_1"/>
<font color="Maroon" title="c10">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E2:4_4_12_1_1_1"><i><font color="Green" title="E30">A48</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:4_4_12_1_1"/><i><font color="Green" title="E29">A49</font></i>: 
<span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_12_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:4_4_12_1_1_2"/><i><font color="Green" title="E30">A50</font></i>: 
<span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E23:4_4"><i><font color="Green" title="E22">A36</font></i></a>, <a class="txt" href="yoneda_1.html#E4:4_4_12_1_1"><i><font color="Green" title="E27">A45</font></i></a>, <a class="ref" href="ens_1.html#D21" title="ENS_1:def.21">ENS_1:def 21</a></i>;<br/>
<a NAME="E2:4_4_12_1_1_2"/><i><font color="Green" title="E31">A51</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E4:4_4_12_1_1"><i><font color="Green" title="E27">A45</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/>
<a NAME="E3:4_4_12_1_1_2"/><b>then </b><i><font color="Green" title="E32">A52</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>;<br/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c2">f</font></span>)</span> = 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c20">x</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:4_4_12_1_1_2"><i><font color="Green" title="E31">A51</font></i></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>
<br/>.= 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E4:4_4_12_1_1"><i><font color="Green" title="E27">A45</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>
;<br/>
<b>hence </b><a NAME="E5:4_4_12_1_1_2"/>
<span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4_12_1_1_2"><i><font color="Green" title="E30">A50</font></i></a>, <a class="txt" href="yoneda_1.html#E3:4_4_12_1_1_2"><i><font color="Green" title="E32">A52</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c21">h</font> = <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E9:4_4_12_1_1"/><i><font color="Green" title="E30">A53</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c20">x</font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E4:4_4_12_1_1"><i><font color="Green" title="E27">A45</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/><a NAME="E10:4_4_12_1_1"/><i><font color="Green" title="E31">A54</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E4:4_4_12_1_1"><i><font color="Green" title="E27">A45</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4_12"><i><font color="Green" title="E25">A43</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c20">x</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E4:4_4_12_1_1"><i><font color="Green" title="E27">A45</font></i></a>, <a class="ref" href="ens_1.html#D20" title="ENS_1:def.20">ENS_1:def 20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c20">x</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c2">f</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E6:4_4_12_1_1"><i><font color="Green" title="E28">A46</font></i></a>, <a class="ref" href="ens_1.html#D21" title="ENS_1:def.21">ENS_1:def 21</a></i>
<br/>.= 
<font color="Maroon" title="c10">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c2">f</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E9:4_4_12_1_1"><i><font color="Green" title="E30">A53</font></i></a>, <a class="txt" href="yoneda_1.html#E10:4_4_12_1_1"><i><font color="Green" title="E31">A54</font></i></a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</a></i>
<br/>.= 
<font color="Maroon" title="c10">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c21">h</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E23:4_4"><i><font color="Green" title="E22">A36</font></i></a>, <a class="txt" href="yoneda_1.html#E4:4_4_12_1_1"><i><font color="Green" title="E27">A45</font></i></a>, <a class="ref" href="ens_1.html#D21" title="ENS_1:def.21">ENS_1:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E7:4_4_12_1_1"><i><font color="Green" title="E29">A49</font></i></a>, <a class="ref" href="ens_1.html#D20" title="ENS_1:def.20">ENS_1:def 20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E24:4_4"><i><font color="Green" title="E23">A37</font></i></a>, <a class="txt" href="yoneda_1.html#E1:4_4_12"><i><font color="Green" title="E25">A43</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><b>hence </b><a NAME="E12:4_4_12_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_12_1_2"><b>suppose </b></a><a NAME="E1:4_4_12_1_2"/><i><font color="Green" title="E26">A55</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:4_4_12_1_2"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4_12"><i><font color="Green" title="E25">A43</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><b>hence </b><a NAME="E3:4_4_12_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E1:4_4_12_1_2"><i><font color="Green" title="E26">A55</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E26:4_4"/><i><font color="Green" title="E25">A56</font></i>: 
(  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c9">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:4"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="yoneda_1.html#E1:4_4"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="cat_1.html#T126" title="CAT_1:th.126">CAT_1:126</a></i>;<br/>
<a NAME="E27:4_4"/><i><font color="Green" title="E26">A57</font></i>: 
(  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:4"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="yoneda_1.html#E1:4_4"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="cat_1.html#T126" title="CAT_1:th.126">CAT_1:126</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span> = 
<font color="Maroon" title="c11">f1</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c12">f2</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:4_4"><i><font color="Green" title="E25">A56</font></i></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c9">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:4_4"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="yoneda_1.html#E7:4_4"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="yoneda_1.html#E8:4_4"><i><font color="Green" title="E12">A12</font></i></a>, <a class="txt" href="yoneda_1.html#E10:4_4"><i><font color="Green" title="E13">A13</font></i></a>, <a class="txt" href="yoneda_1.html#E11:4_4"><i><font color="Green" title="E14">A15</font></i></a>, <a class="ref" href="yoneda_1.html#T1" target="_self" title="YONEDA_1:th.1">Th1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c10">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E23:4_4"><i><font color="Green" title="E22">A36</font></i></a>, <a class="txt" href="yoneda_1.html#E24:4_4"><i><font color="Green" title="E23">A37</font></i></a>, <a class="txt" href="yoneda_1.html#E25:4_4"><i><font color="Green" title="E24">A42</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>
<br/>.= 
<font color="Maroon" title="c15">f3</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c16">f4</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E14:4_4"><i><font color="Green" title="E15">A16</font></i></a>, <a class="txt" href="yoneda_1.html#E16:4_4"><i><font color="Green" title="E16">A17</font></i></a>, <a class="txt" href="yoneda_1.html#E17:4_4"><i><font color="Green" title="E17">A19</font></i></a>, <a class="txt" href="yoneda_1.html#E19:4_4"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="yoneda_1.html#E20:4_4"><i><font color="Green" title="E19">A22</font></i></a>, <a class="ref" href="yoneda_1.html#T1" target="_self" title="YONEDA_1:th.1">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E27:4_4"><i><font color="Green" title="E26">A57</font></i></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></i>
;<br/>
<b>hence </b><a NAME="E29:4_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:4"/>
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E3:4"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a></i>;<br/>


</div>
