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

<b>set </b><font color="Maroon" title="c2">X</font> =  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font>,   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font>] <b>means </b> <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> $2;<br/>
<b>let </b><font color="Maroon" title="c3">c1</font>, <font color="Maroon" title="c4">c2</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:87_1_2"/><i><font color="Green" title="E48">A4</font></i>: 
for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">c1</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">f</font>] )
 <b>and </b><br/><a NAME="E2:87_1_2"/><i><font color="Green" title="E49">A5</font></i>: 
for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">c1</font> holds <br/><font color="Maroon" title="c3">c1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Olive" title="b2">f</font>
 <b>and </b><br/><a NAME="E3:87_1_2"/><i><font color="Green" title="E50">A6</font></i>: 
for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">c2</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">f</font>] )
 <b>and </b><br/><a NAME="E4:87_1_2"/><i><font color="Green" title="E51">A7</font></i>: 
for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">c2</font> holds <br/><font color="Maroon" title="c4">c2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Olive" title="b2">f</font>
 ;<br/>

<a NAME="E5:87_1_2"/><i><font color="Green" title="E52">A8</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">c1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">c2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E6:87_1_2"/><i><font color="Green" title="E53">A9</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">c2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:87_1_2_1_1"/><i><font color="Green" title="E54">A10</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">c1</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">g</font>, <font color="Maroon" title="c7">f</font> being    <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:87_1_2_1_1"/><i><font color="Green" title="E55">A11</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c7">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="grcat_1.html#E5:87_1_2"><i><font color="Green" title="E52">A8</font></i></a>, <a class="ref" href="grcat_1.html#T2" target="_self" title="GRCAT_1:th.2">Th2</a></i>;<br/><a NAME="E4:87_1_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c7">f</font>]
 <b>by </b><i><a class="txt" href="grcat_1.html#E1:87_1_2"><i><font color="Green" title="E48">A4</font></i></a>, <a class="txt" href="grcat_1.html#E1:87_1_2_1_1"><i><font color="Green" title="E54">A10</font></i></a>, <a class="txt" href="grcat_1.html#E3:87_1_2_1_1"><i><font color="Green" title="E55">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E5:87_1_2_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">c2</font>
 <b>by </b><i><a class="txt" href="grcat_1.html#E3:87_1_2"><i><font color="Green" title="E50">A6</font></i></a>, <a class="txt" href="grcat_1.html#E3:87_1_2_1_1"><i><font color="Green" title="E55">A11</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:87_1_2_1"/><b>then </b><i><font color="Green" title="E54">A12</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">c1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">c2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:87_1_2_1_2"/><i><font color="Green" title="E55">A13</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">c2</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">g</font>, <font color="Maroon" title="c7">f</font> being    <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:87_1_2_1_2"/><i><font color="Green" title="E56">A14</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c7">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="grcat_1.html#E5:87_1_2"><i><font color="Green" title="E52">A8</font></i></a>, <a class="ref" href="grcat_1.html#T2" target="_self" title="GRCAT_1:th.2">Th2</a></i>;<br/><a NAME="E4:87_1_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c7">f</font>]
 <b>by </b><i><a class="txt" href="grcat_1.html#E3:87_1_2"><i><font color="Green" title="E50">A6</font></i></a>, <a class="txt" href="grcat_1.html#E1:87_1_2_1_2"><i><font color="Green" title="E55">A13</font></i></a>, <a class="txt" href="grcat_1.html#E3:87_1_2_1_2"><i><font color="Green" title="E56">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E5:87_1_2_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">c1</font>
 <b>by </b><i><a class="txt" href="grcat_1.html#E1:87_1_2"><i><font color="Green" title="E48">A4</font></i></a>, <a class="txt" href="grcat_1.html#E3:87_1_2_1_2"><i><font color="Green" title="E56">A14</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:87_1_2_1"/><b>then </b>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">c2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">c1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:87_1_2_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">c2</font>
 <b>by </b><i><a class="txt" href="grcat_1.html#E2:87_1_2_1"><i><font color="Green" title="E54">A12</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c5">V'</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">c1</font>;<br/>
<a NAME="E7:87_1_2"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">c1</font> holds <br/><font color="Maroon" title="c3">c1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:87_1_2_2"/><i><font color="Green" title="E54">A15</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">c1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c7">y</font> as    <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="grcat_1.html#E5:87_1_2"><i><font color="Green" title="E52">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:87_1_2_2"/>
<font color="Maroon" title="c3">c1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Maroon" title="c9">y</font>
 
<b>by </b><i><a class="txt" href="grcat_1.html#E2:87_1_2"><i><font color="Green" title="E49">A5</font></i></a>, <a class="txt" href="grcat_1.html#E1:87_1_2_2"><i><font color="Green" title="E54">A15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:87_1_2_2"/>
<font color="Maroon" title="c3">c1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="grcat_1.html#E4:87_1_2"><i><font color="Green" title="E51">A7</font></i></a>, <a class="txt" href="grcat_1.html#E6:87_1_2"><i><font color="Green" title="E53">A9</font></i></a>, <a class="txt" href="grcat_1.html#E1:87_1_2_2"><i><font color="Green" title="E54">A15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:87_1_2"/>
<font color="Maroon" title="c3">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c2</font>
 <b>by </b><i><a class="txt" href="grcat_1.html#E6:87_1_2"><i><font color="Green" title="E53">A9</font></i></a>, <a class="ref" href="binop_1.html#T32" title="BINOP_1:th.32">BINOP_1:32</a></i>;<br/>


</div>
