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

<b>set </b><font color="Maroon">X</font> = the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">C</font></span><a href="cat_2.html#K11">:]</a></span>;<br/>
<b>set </b><font color="Maroon">Y</font> =  <a href="ens_1.html#K2" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" target="_self">Hom</a> <font color="Maroon">C</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Olive">f</font>,<font color="Olive">g</font></span><a href="cat_2.html#K13">]</a></span> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <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> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Olive">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Olive">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="Olive">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Olive">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#K22" target="_self">hom</a> <font color="Olive">f</font>,<font color="Olive">g</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>;<br/>
<a NAME="E1:82_1_1"/><i><font color="Green">E16</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> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">C</font></span><a href="cat_2.html#K11">:]</a></span> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K2" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" target="_self">Hom</a> <font color="Maroon">C</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1_1"><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="E1:82_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">C</font></span><a href="cat_2.html#K11">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>, <font color="Maroon">g</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:82_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_2.html#K13">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>take </b><font color="Maroon">y</font> = <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> <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>,<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#K22" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>;<br/>

<a NAME="E4:82_1_1_1"/>
<font color="Maroon">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" target="_self">Hom</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T16" target="_self">Th16</a></i>;<br/>
<b>hence </b><a NAME="E5:82_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K2" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" target="_self">Hom</a> <font color="Maroon">C</font></span>)</span>
 ;<br/>

<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>let </b><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="E6:82_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">f'</font>,<font color="Maroon">g'</font></span><a href="cat_2.html#K13">]</a></span>
 ;<br/>

<a NAME="E7:82_1_1_1"/><b>then </b>
( <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 
<b>by </b><i><a class="ref" href="ens_1.html#T19" target="_self">Th19</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E8:82_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <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> <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>,<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#K22" target="_self">hom</a> <font color="Maroon">f'</font>,<font color="Maroon">g'</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:82_1_1"/><i><font color="Green">E19</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">C</font></span><a href="cat_2.html#K11">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a>  <a href="ens_1.html#K2" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" target="_self">Hom</a> <font color="Maroon">C</font></span>)</span> )
 <b>and </b><br/><a NAME="E4:82_1_1"/><i><font color="Green">E20</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> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">C</font></span><a href="cat_2.html#K11">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">C</font></span><a href="cat_2.html#K11">:]</a></span>, <a href="ens_1.html#K2" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" target="_self">Hom</a> <font color="Maroon">C</font></span>)</span> <b>by </b><i><a class="ref" href="ens_1.html#T19" target="_self">Th19</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">h</font>
;<br/>

<b>thus </b><a NAME="E6:82_1_1"/>
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> holds  <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Olive">f</font>,<font color="Olive">g</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R1">=</a> <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> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Olive">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Olive">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="Olive">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Olive">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#K22" target="_self">hom</a> <font color="Olive">f</font>,<font color="Olive">g</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="ens_1.html#T20" target="_self">Th20</a></i>;<br/>


</div>
