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

<b>set </b><font color="Maroon">F1</font> = <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">F2</font> = <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>;<br/>
<a NAME="E1:5_1_1"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">o</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font> holds  <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Olive">o</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Olive">o</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">o</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Olive">o</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Olive">o</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E1:5_1_1_1"/><i><font color="Green">E18</font></i>: 
 <a href="ens_1.html#K22">hom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="ens_1.html#T53">ENS_1:53</a></i>;<br/>
<a NAME="E2:5_1_1_1"/><i><font color="Green">E21</font></i>: 
 <a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K2">Maps</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K10">fComp</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K11">fId</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="ens_1.html#D14">ENS_1:def 14</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">m</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> as   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i>, <a class="ref" href="ens_1.html#T48">ENS_1:48</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m'</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i>, <a class="ref" href="ens_1.html#T48">ENS_1:48</a></i>;<br/>
<i><font color="Green">E22</font></i>:  <a href="cat_1.html#K2">dom</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#D2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="ens_1.html#K3">dom</a> <font color="Maroon">m'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D10">ENS_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D4">ENS_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K24">hom?-</a> <span class="p3">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="ens_1.html#T60">ENS_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K24">hom?-</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="cat_1.html#D20">CAT_1:def 20</a></i>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="ens_1.html#D26">ENS_1:def 26</a></i>
;<br/>
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#D3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="ens_1.html#K4">cod</a> <font color="Maroon">m'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D11">ENS_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D5">ENS_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K24">hom?-</a> <span class="p3">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="ens_1.html#T60">ENS_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K24">hom?-</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="cat_1.html#D20">CAT_1:def 20</a></i>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="ens_1.html#D26">ENS_1:def 26</a></i>
;<br/>
<b>hence </b><a NAME="E7:5_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">o</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E2:5_1_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">o</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">o</font>) <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E1:5_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">o</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:5_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">o</font>) <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">t</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font>,the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E4:5_1_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">o</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font> holds  <font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">o</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S8">FUNCT_2:sch 8</a>();<br/></i>
<a NAME="E5:5_1_1"/><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:5_1_1"/>
for <font color="Olive">o</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>  holds  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Olive">o</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Olive">o</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E7:5_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#R1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/>
<a NAME="E8:5_1_1"/>
for <font color="Olive">o</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font> holds  <font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> is    <a href="cat_1.html#M1">Morphism</a> of <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Olive">o</font>,<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Olive">o</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E1:5_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">o</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:5_1_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> is    <a href="cat_1.html#M1">Morphism</a> of <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">o</font>,<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E3:5_1_1_3"/>
<font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> is    <a href="cat_1.html#M1">Morphism</a> of <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">o</font>,<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">o</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as    <a href="nattra_1.html#M1">transformation</a> of <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>,<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <b>by </b><i>, <a class="ref" href="nattra_1.html#D3">NATTRA_1:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">B</font> =  <a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">A</font>;<br/>
<a NAME="E10:5_1_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>  st  <a href="cat_1.html#K6">Hom</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">g</font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">a</font>,<font color="Olive">b</font> holds  <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Olive">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Olive">g</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Olive">a</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:5_1_1_4"/><i><font color="Green">E39</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">g</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>

<a NAME="E2:5_1_1_4"/><i><font color="Green">E41</font></i>: 
 <a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K2">Maps</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K10">fComp</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K11">fId</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="ens_1.html#D14">ENS_1:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b</font> as   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">A</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">f2</font> = <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font> as   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">A</font></span>)</span> ;<br/>
<i><font color="Green">E42</font></i>: <font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b</font> = 
<font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#T53">ENS_1:53</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">f1'</font> = <font color="Maroon">f1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="ref" href="ens_1.html#T48">ENS_1:48</a></i>;<br/>
<a NAME="E7:5_1_1_4"/><i><font color="Green">E43</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_4_2"><b>proof </b></a><div class="add">

<i><font color="Green">E45</font></i>:  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f1</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f1</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#D2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="ens_1.html#K3">dom</a> <font color="Maroon">f1'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D10">ENS_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D4">ENS_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">f1</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f1</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#D3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="ens_1.html#K4">cod</a> <font color="Maroon">f1'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D11">ENS_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D5">ENS_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<b>hence </b><a NAME="E3:5_1_1_4_2"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="yoneda_1.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E46</font></i>: <font color="Maroon">f2</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K20">hom?-</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#D22">ENS_1:def 22</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">f2'</font> = <font color="Maroon">f2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="ref" href="ens_1.html#T47">ENS_1:47</a></i>;<br/>
<a NAME="E10:5_1_1_4"/><i><font color="Green">E47</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_4_4"><b>proof </b></a><div class="add">

<i><font color="Green">E50</font></i>:  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f2</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#D2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="ens_1.html#K3">dom</a> <font color="Maroon">f2'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D10">ENS_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D4">ENS_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="yoneda_1.html#T1" target="_self">Th1</a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f2</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#D3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="ens_1.html#K4">cod</a> <font color="Maroon">f2'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D11">ENS_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D5">ENS_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="yoneda_1.html#T1" target="_self">Th1</a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<b>hence </b><a NAME="E3:5_1_1_4_4"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E11:5_1_1_4"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i>, , <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f3</font> = <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font> as   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">A</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">f4</font> = <font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font> as   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">A</font></span>)</span> ;<br/>
<i><font color="Green">E53</font></i>: <font color="Maroon">f3</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K20">hom?-</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#D22">ENS_1:def 22</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">f3'</font> = <font color="Maroon">f3</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="ref" href="ens_1.html#T47">ENS_1:47</a></i>;<br/>
<a NAME="E16:5_1_1_4"/><i><font color="Green">E54</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f3</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f3</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_4_6"><b>proof </b></a><div class="add">

<i><font color="Green">E56</font></i>:  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f3</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f3</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#D2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="ens_1.html#K3">dom</a> <font color="Maroon">f3'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D10">ENS_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f3</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D4">ENS_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">f3</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f3</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#D3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="ens_1.html#K4">cod</a> <font color="Maroon">f3'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D11">ENS_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f3</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D5">ENS_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<b>hence </b><a NAME="E3:5_1_1_4_6"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f3</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f3</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E57</font></i>: <font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#T53">ENS_1:53</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">f4'</font> = <font color="Maroon">f4</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="ref" href="ens_1.html#T48">ENS_1:48</a></i>;<br/>
<a NAME="E19:5_1_1_4"/><i><font color="Green">E58</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f4</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f4</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_4_8"><b>proof </b></a><div class="add">

<i><font color="Green">E59</font></i>:  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f4</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f4</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#D2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="ens_1.html#K3">dom</a> <font color="Maroon">f4'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D10">ENS_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f4</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D4">ENS_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">f4</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f4</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#D3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="ens_1.html#K4">cod</a> <font color="Maroon">f4'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D11">ENS_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f4</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D5">ENS_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<b>hence </b><a NAME="E3:5_1_1_4_8"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f4</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f4</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E20:5_1_1_4"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">f4</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f3</font>
 
<b>by </b><i>, <a class="txt" href="yoneda_1.html#E1:5_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<a NAME="E21:5_1_1_4"/><i><font color="Green">E61</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_4_9"><b>proof </b></a><div class="add">

<a NAME="E1:5_1_1_4_9"/><i><font color="Green">E62</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_4_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_1_4_9_1"/>
(  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_4_9_1_1"><b>suppose </b></a><a NAME="E1:5_1_1_4_9_1_1"/>
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:5_1_1_4_9_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="ens_1.html#T42">ENS_1:42</a></i>;<br/><a NAME="E3:5_1_1_4_9_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><b>hence </b><a NAME="E4:5_1_1_4_9_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="yoneda_1.html#E2:5_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_4_9_1_2"><b>suppose </b></a><a NAME="E1:5_1_1_4_9_1_2"/><i><font color="Green">E65</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/><b>assume </b><a NAME="E2:5_1_1_4_9_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span>
 ;<br/><a NAME="E3:5_1_1_4_9_1_2"/><i><font color="Green">E67</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/><a NAME="E4:5_1_1_4_9_1_2"/><i><font color="Green">E68</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E5:5_1_1_4_9_1_2"/>
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E2:5_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="yoneda_1.html#E5:5_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E6:5_1_1_4_9_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E4:5_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yoneda_1.html#E7:5_1_1"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E22:5_1_1_4"/><i><font color="Green">E69</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_4_10"><b>proof </b></a><div class="add">

<a NAME="E1:5_1_1_4_10"/><i><font color="Green">E70</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_4_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_1_4_10_1"/>
(  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_4_10_1_1"><b>suppose </b></a><a NAME="E1:5_1_1_4_10_1_1"/>
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:5_1_1_4_10_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="ens_1.html#T42">ENS_1:42</a></i>;<br/><a NAME="E3:5_1_1_4_10_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><b>hence </b><a NAME="E4:5_1_1_4_10_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E2:5_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="yoneda_1.html#E4:5_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_4_10_1_2"><b>suppose </b></a><a NAME="E1:5_1_1_4_10_1_2"/><i><font color="Green">E72</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/><b>assume </b><a NAME="E2:5_1_1_4_10_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span>
 ;<br/><a NAME="E3:5_1_1_4_10_1_2"/><i><font color="Green">E74</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E4:5_1_1_4_10_1_2"/>
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E2:5_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="yoneda_1.html#E4:5_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E5:5_1_1_4_10_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E5:5_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="yoneda_1.html#E7:5_1_1"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E23:5_1_1_4"/><i><font color="Green">E75</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<a NAME="E24:5_1_1_4"/><i><font color="Green">E76</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_4_11"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_4_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_1_4_11_1"/>
(  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_4_11_1_1"><b>suppose </b></a><a NAME="E1:5_1_1_4_11_1_1"/><i><font color="Green">E77</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"> <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
<br/>.= 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:5_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="yoneda_1.html#E5:5_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:5_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="yoneda_1.html#E5:5_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
;<br/><b>hence </b><a NAME="E3:5_1_1_4_11_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_4_11_1_2"><b>suppose </b></a><a NAME="E1:5_1_1_4_11_1_2"/><i><font color="Green">E78</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:5_1_1_4_11_1_2"/><b>then </b><i><font color="Green">E79</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="ens_1.html#T42">ENS_1:42</a></i>;<br/><a NAME="E3:5_1_1_4_11_1_2"/><i><font color="Green">E81</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="yoneda_1.html#E2:5_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="yoneda_1.html#E5:5_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="ens_1.html#T42">ENS_1:42</a></i>;<br/> <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
<br/>.= 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:5_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="yoneda_1.html#E7:5_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="yoneda_1.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
;<br/><b>hence </b><a NAME="E5:5_1_1_4_11_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E25:5_1_1_4"/><i><font color="Green">E82</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">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">hom</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_4_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:5_1_1_4_12"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_4_12_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_1_4_12_1"/>
(  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_4_12_1_1"><b>suppose </b></a><a NAME="E1:5_1_1_4_12_1_1"/>
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:5_1_1_4_12_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="ens_1.html#T42">ENS_1:42</a></i>;<br/><a NAME="E3:5_1_1_4_12_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E7:5_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E4:5_1_1_4_12_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="yoneda_1.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font> ;<br/><a NAME="E6:5_1_1_4_12_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_4_12_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:5_1_1_4_12_1_1_1"/><i><font color="Green">E87</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="yoneda_1.html#T3" target="_self">Th3</a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>
<a NAME="E2:5_1_1_4_12_1_1_1"/><i><font color="Green">E88</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<i><font color="Green">E89</font></i>:  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> = 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="yoneda_1.html#T3" target="_self">Th3</a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>
;<br/>
<a NAME="E4:5_1_1_4_12_1_1_1"/>
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>;<br/>
<b>hence </b><a NAME="E5:5_1_1_4_12_1_1_1"/>
<font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:5_1_1_4_12_1_1"/><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_4_12_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:5_1_1_4_12_1_1_2"/><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:5_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="yoneda_1.html#T3" target="_self">Th3</a>, <a class="ref" href="ens_1.html#D21">ENS_1:def 21</a></i>;<br/>
<a NAME="E2:5_1_1_4_12_1_1_2"/><i><font color="Green">E93</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="yoneda_1.html#T3" target="_self">Th3</a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>
<a NAME="E3:5_1_1_4_12_1_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>;<br/>
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> = 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>
<br/>.= 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="yoneda_1.html#T3" target="_self">Th3</a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>
;<br/>
<b>hence </b><a NAME="E5:5_1_1_4_12_1_1_2"/>
<span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">h</font> = <span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font> ;<br/><a NAME="E9:5_1_1_4_12_1_1"/><i><font color="Green">E95</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yoneda_1.html#T3" target="_self">Th3</a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/><a NAME="E10:5_1_1_4_12_1_1"/><i><font color="Green">E96</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="yoneda_1.html#T3" target="_self">Th3</a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E7:5_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="yoneda_1.html#T3" target="_self">Th3</a>, <a class="ref" href="ens_1.html#D20">ENS_1:def 20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="ens_1.html#D21">ENS_1:def 21</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="cat_1.html#T43">CAT_1:43</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:5_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="yoneda_1.html#T3" target="_self">Th3</a>, <a class="ref" href="ens_1.html#D21">ENS_1:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="ens_1.html#D20">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">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E4:5_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yoneda_1.html#E7:5_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/><b>hence </b><a NAME="E12:5_1_1_4_12_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_4_12_1_2"><b>suppose </b></a><a NAME="E1:5_1_1_4_12_1_2"/><i><font color="Green">E98</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:5_1_1_4_12_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E7:5_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><b>hence </b><a NAME="E3:5_1_1_4_12_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yoneda_1.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E26:5_1_1_4"/><i><font color="Green">E99</font></i>: 
(  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, , <a class="ref" href="cat_1.html#T126">CAT_1:126</a></i>;<br/>
<a NAME="E27:5_1_1_4"/><i><font color="Green">E100</font></i>: 
(  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, , <a class="ref" href="cat_1.html#T126">CAT_1:126</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font></span>)</span> = 
<font color="Maroon">f1</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f2</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E7:5_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="yoneda_1.html#T1" target="_self">Th1</a>, , , </i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K18">hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:5_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="yoneda_1.html#E4:5_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yoneda_1.html#E5:5_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>
<br/>.= 
<font color="Maroon">f3</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f4</font>
<b>by </b><i>, <a class="txt" href="yoneda_1.html#E1:5_1_1"><i><font color="Green">E17</font></i></a>, , , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="yoneda_1.html#T2" target="_self">Th2</a>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
;<br/>
<b>hence </b><a NAME="E29:5_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as    <a href="nattra_1.html#M2">natural_transformation</a> of <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>,<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <b>by </b><i>, <a class="ref" href="nattra_1.html#D8">NATTRA_1:def 8</a></i>;<br/>
<a NAME="E12:5_1_1"/>
for <font color="Olive">o</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font> holds  <font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Olive">o</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Olive">o</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">o</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/>

<font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">o</font> = 
<font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:5_1_1_5"/>
<font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">o</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:5_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="nattra_1.html#M2">natural_transformation</a> of <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>,<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> st <br/>for <font color="Olive">o</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font> holds  <font color="Olive">b<sub>1</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Olive">o</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Olive">o</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">o</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 ;<br/>


</div>
