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

<b>set </b><font color="Maroon">X</font> =  <a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="grcat_1.html#M3" target="_self">Element</a> of  <a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font>,   <a href="grcat_1.html#M3" target="_self">Element</a> of  <a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font>] <b>means </b> <a href="grcat_1.html#K21" target="_self">dom</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#K22" target="_self">cod</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>let </b><font color="Maroon">c1</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font>, <font color="Maroon">c2</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:91_1_2"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">g</font>, <font color="Olive">f</font> being   <a href="grcat_1.html#M3" target="_self">Element</a> of  <a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c1</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">g</font>,<font color="Olive">f</font>] )
 <b>and </b><br/><a NAME="E2:91_1_2"/><i><font color="Green">E90</font></i>: 
for <font color="Olive">g</font>, <font color="Olive">f</font> being   <a href="grcat_1.html#M3" target="_self">Element</a> of  <a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font>  st <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c1</font> holds <br/><font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="grcat_1.html#K15" target="_self">*</a> <font color="Olive">f</font>
 <b>and </b><br/><a NAME="E3:91_1_2"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">g</font>, <font color="Olive">f</font> being   <a href="grcat_1.html#M3" target="_self">Element</a> of  <a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c2</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">g</font>,<font color="Olive">f</font>] )
 <b>and </b><br/><a NAME="E4:91_1_2"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">g</font>, <font color="Olive">f</font> being   <a href="grcat_1.html#M3" target="_self">Element</a> of  <a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font>  st <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c2</font> holds <br/><font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="grcat_1.html#K15" target="_self">*</a> <font color="Olive">f</font>
 ;<br/>

<a NAME="E5:91_1_2"/><i><font color="Green">E96</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c1</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c2</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E6:91_1_2"/><i><font color="Green">E97</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_1_1"><b>now </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="E1:91_1_2_1_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c1</font>
 ;<br/><b>then consider </b><font color="Maroon">g</font> being    <a href="grcat_1.html#M3" target="_self">Element</a> of  <a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font>, <font color="Maroon">f</font> being    <a href="grcat_1.html#M3" target="_self">Element</a> of  <a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:91_1_2_1_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="grcat_1.html#T12" target="_self">Th12</a>, </i>;<br/><a NAME="E4:91_1_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">g</font>,<font color="Maroon">f</font>]
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E5:91_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c2</font>
 <b>by </b><i><a class="ref" href="grcat_1.html#T9" target="_self">Th9</a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E2:91_1_2_1"/><b>then </b><i><font color="Green">E102</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_1_2"><b>now </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="E1:91_1_2_1_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c2</font>
 ;<br/><b>then consider </b><font color="Maroon">g</font> being    <a href="grcat_1.html#M3" target="_self">Element</a> of  <a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font>, <font color="Maroon">f</font> being    <a href="grcat_1.html#M3" target="_self">Element</a> of  <a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:91_1_2_1_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="grcat_1.html#T12" target="_self">Th12</a>, </i>;<br/><a NAME="E4:91_1_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">g</font>,<font color="Maroon">f</font>]
 <b>by </b><i><a class="ref" href="grcat_1.html#T9" target="_self">Th9</a>, , </i>;<br/><b>hence </b><a NAME="E5:91_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c1</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E4:91_1_2_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:91_1_2_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">V'</font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c1</font>;<br/>
<a NAME="E7:91_1_2"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c1</font> holds <br/><font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_2"><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>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:91_1_2_2"/><i><font color="Green">E168</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="grcat_1.html#M3" target="_self">Element</a> of  <a href="grcat_1.html#K20" target="_self">Morphs</a> <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="grcat_1.html#T12" target="_self">Th12</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:91_1_2_2"/>
<font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="grcat_1.html#K15" target="_self">*</a> <font color="Maroon">y</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:91_1_2_2"/>
<font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="grcat_1.html#T11" target="_self">Th11</a>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:91_1_2"/>
<font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font>
 <b>by </b><i>, <a class="ref" href="partfun1.html#T35">PARTFUN1:35</a></i>;<br/>


</div>
