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

<b>let </b><font color="Maroon">V</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/>





<b>assume </b><a NAME="E1:73"/><i><font color="Green">E16</font></i>: 
 <a href="ens_1.html#K17" target="_self">Hom</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">T</font> =  <a href="ens_1.html#K20" target="_self">hom?-</a> <font color="Maroon">a</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" target="_self">Ens</a> <font color="Maroon">V</font></span>)</span> <b>by </b><i><a class="ref" href="ens_1.html#D7" target="_self">Def7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:73_1"/>
for <font color="Olive">c</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>  ex <font color="Olive">d</font> being  <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" target="_self">Ens</a> <font color="Maroon">V</font></span>)</span> st <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Olive">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:73_1_1"/>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K17" target="_self">Hom</a> <font color="Maroon">C</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">A</font> =  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">d</font> =  <a href="ens_1.html#K13" target="_self">@</a> <font color="Maroon">A</font>;<br/>

<b>thus </b><a NAME="E3:73_1_1"/>
<font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="ens_1.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:73_1"/>
for <font color="Olive">f</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> holds <br/> ( <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Olive">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font></span>)</span></span>)</span> &amp; <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Olive">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/>

<b>set </b><font color="Maroon">b</font> =  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">c</font> =  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>;<br/>
<a NAME="E1:73_1_2"/>
(  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K17" target="_self">Hom</a> <font color="Maroon">C</font> &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K17" target="_self">Hom</a> <font color="Maroon">C</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">A</font> =  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<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="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>;<br/>
<i><font color="Green">E17</font></i>: <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" target="_self">hom</a> <font color="Maroon">a</font>,<font color="Maroon">f</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> = 
 <a href="ens_1.html#K16" target="_self">@</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="ens_1.html#T10" target="_self">Th10</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </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="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </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="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="cat_2.html#K12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i/>
;<br/>
<b>thus </b><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</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="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="ens_1.html#K13" target="_self">@</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i>, <a class="ref" href="ens_1.html#T12" target="_self">Th12</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>

<b>thus </b><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</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="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="ens_1.html#K13" target="_self">@</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i>, <a class="ref" href="ens_1.html#T12" target="_self">Th12</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>, <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/><b>assume </b><a NAME="E3:73_1"/><i><font color="Green">E19</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">f</font>
 ;<br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" target="_self">hom</a> <font color="Maroon">a</font>,<font color="Maroon">f</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> = 
 <a href="ens_1.html#K16" target="_self">@</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="ens_1.html#T10" target="_self">Th10</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </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="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </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="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="cat_2.html#K12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i/>
;<br/><a NAME="E5:73_1"/><b>then </b><i><font color="Green">E20</font></i>: 
( <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K18" target="_self">hom</a> <font color="Maroon">a</font>,<font color="Maroon">f</font> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E6:73_1"/><b>then </b><i><font color="Green">E62</font></i>: 
(  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i/>;<br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<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> <font color="Maroon">a</font>,<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="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" target="_self">hom</a> <font color="Maroon">a</font>,<font color="Maroon">g</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> = 
 <a href="ens_1.html#K16" target="_self">@</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="ens_1.html#T10" target="_self">Th10</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </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="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </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="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="cat_2.html#K12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i/>
;<br/><a NAME="E8:73_1"/><b>then </b><i><font color="Green">E63</font></i>: 
( <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K18" target="_self">hom</a> <font color="Maroon">a</font>,<font color="Maroon">g</font> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<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> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<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="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E9:73_1"/><b>then </b><i><font color="Green">E74</font></i>: 
(  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<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/><a NAME="E10:73_1"/>
(  <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">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> &amp;  <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">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> )
 <b>by </b><i>, <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>;<br/><b>hence </b><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<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="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" target="_self">hom</a> <font color="Maroon">a</font>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#T10" target="_self">Th10</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<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="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K18" target="_self">hom</a> <font color="Maroon">a</font>,<font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="ens_1.html#K18" target="_self">hom</a> <font color="Maroon">a</font>,<font color="Maroon">f</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, <a class="ref" href="ens_1.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="ens_1.html#T15" target="_self">Th15</a>, <a class="ref" href="ens_1.html#T16" target="_self">Th16</a>, <a class="ref" href="ens_1.html#T17" target="_self">Th17</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, <a class="ref" href="ens_1.html#T15" target="_self">Th15</a>, <a class="ref" href="ens_1.html#T17" target="_self">Th17</a>, <a class="ref" href="ens_1.html#T5" target="_self">Th5</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:73"/>
 <a href="ens_1.html#K20" target="_self">hom?-</a> <font color="Maroon">a</font> is    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">C</font>, <a href="ens_1.html#K12" target="_self">Ens</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T96">CAT_1:96</a></i>;<br/>


</div>
