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<b>set </b><font color="Maroon" title="c3">X</font> =  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">V</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">V</font>,   <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">V</font>] <b>means </b> <a href="modcat_1.html#K5" title="MODCAT_1:func.5">dom'</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modcat_1.html#K6" title="MODCAT_1:func.6">cod'</a> $2;<br/>
<b>let </b><font color="Maroon" title="c4">c1</font>, <font color="Maroon" title="c5">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="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">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">c1</font> iff  <a href="modcat_1.html#K5" title="MODCAT_1:func.5">dom'</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modcat_1.html#K6" title="MODCAT_1:func.6">cod'</a> <font color="Olive" title="b2">f</font> ) ) &amp; ( for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">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">c1</font> holds <br/><font color="Maroon" title="c4">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="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Olive" title="b2">f</font> ) &amp; ( for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">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="c5">c2</font> iff  <a href="modcat_1.html#K5" title="MODCAT_1:func.5">dom'</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modcat_1.html#K6" title="MODCAT_1:func.6">cod'</a> <font color="Olive" title="b2">f</font> ) ) &amp; ( for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">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="c5">c2</font> holds <br/><font color="Maroon" title="c5">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="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Olive" title="b2">f</font> ) implies <font color="Maroon" title="c4">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:28_1_2"/><i><font color="Green" title="E20">A4</font></i>: 
for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">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">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="E2:28_1_2"/><i><font color="Green" title="E21">A5</font></i>: 
for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">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">c1</font> holds <br/><font color="Maroon" title="c4">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="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Olive" title="b2">f</font>
 <b>and </b><br/><a NAME="E3:28_1_2"/><i><font color="Green" title="E22">A6</font></i>: 
for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">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="c5">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="E4:28_1_2"/><i><font color="Green" title="E23">A7</font></i>: 
for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">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="c5">c2</font> holds <br/><font color="Maroon" title="c5">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="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Olive" title="b2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c2</font></span><br/>

<a NAME="E5:28_1_2"/><i><font color="Green" title="E24">A8</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">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="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">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="c5">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="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">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="E6:28_1_2"/><i><font color="Green" title="E25">A9</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">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="c5">c2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">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">c1</font> implies <font color="Maroon" title="c6">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="c5">c2</font> )</span><br/><b>assume </b><a NAME="E1:28_1_2_1_1"/><i><font color="Green" title="E26">A10</font></i>: 
<font color="Maroon" title="c6">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">c1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">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="c5">c2</font></span><br/><b>then consider </b><font color="Maroon" title="c7">g</font>, <font color="Maroon" title="c8">f</font> being    <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E3:28_1_2_1_1"/><i><font color="Green" title="E27">A11</font></i>: 
<font color="Maroon" title="c6">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="c7">g</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="modcat_1.html#E5:28_1_2"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="grcat_1.html#T2" title="GRCAT_1:th.2">GRCAT_1:2</a></i>;<br/><a NAME="E4:28_1_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c8">f</font>]
 <b>by </b><i><a class="txt" href="modcat_1.html#E1:28_1_2"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="modcat_1.html#E1:28_1_2_1_1"><i><font color="Green" title="E26">A10</font></i></a>, <a class="txt" href="modcat_1.html#E3:28_1_2_1_1"><i><font color="Green" title="E27">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E5:28_1_2_1_1"/>
<font color="Maroon" title="c6">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="c5">c2</font>
 <b>by </b><i><a class="txt" href="modcat_1.html#E3:28_1_2"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="modcat_1.html#E3:28_1_2_1_1"><i><font color="Green" title="E27">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E2:28_1_2_1"/><b>then </b><i><font color="Green" title="E26">A12</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">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="c5">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="28_1_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">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="c5">c2</font> implies <font color="Maroon" title="c6">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">c1</font> )</span><br/><b>assume </b><a NAME="E1:28_1_2_1_2"/><i><font color="Green" title="E27">A13</font></i>: 
<font color="Maroon" title="c6">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="c5">c2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">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">c1</font></span><br/><b>then consider </b><font color="Maroon" title="c7">g</font>, <font color="Maroon" title="c8">f</font> being    <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E3:28_1_2_1_2"/><i><font color="Green" title="E28">A14</font></i>: 
<font color="Maroon" title="c6">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="c7">g</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="modcat_1.html#E5:28_1_2"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="grcat_1.html#T2" title="GRCAT_1:th.2">GRCAT_1:2</a></i>;<br/><a NAME="E4:28_1_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c8">f</font>]
 <b>by </b><i><a class="txt" href="modcat_1.html#E3:28_1_2"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="modcat_1.html#E1:28_1_2_1_2"><i><font color="Green" title="E27">A13</font></i></a>, <a class="txt" href="modcat_1.html#E3:28_1_2_1_2"><i><font color="Green" title="E28">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E5:28_1_2_1_2"/>
<font color="Maroon" title="c6">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">c1</font>
 <b>by </b><i><a class="txt" href="modcat_1.html#E1:28_1_2"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="modcat_1.html#E3:28_1_2_1_2"><i><font color="Green" title="E28">A14</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E4:28_1_2_1"/><b>then </b>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">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="c4">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="E5:28_1_2_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">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="c5">c2</font>
 <b>by </b><i><a class="txt" href="modcat_1.html#E2:28_1_2_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c6">V0</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">c1</font>;<br/>
<a NAME="E7:28_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="c4">c1</font> holds <br/><font color="Maroon" title="c4">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="c5">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="28_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">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="c4">c1</font> implies <font color="Maroon" title="c4">c1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font> )</span><br/>



<b>assume </b><a NAME="E1:28_1_2_2"/><i><font color="Green" title="E26">A15</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">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="c4">c1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">c1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c8">y</font> as    <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="modcat_1.html#E5:28_1_2"><i><font color="Green" title="E24">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="E3:28_1_2_2"/>
<font color="Maroon" title="c4">c1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c10">y</font>
 
<b>by </b><i><a class="txt" href="modcat_1.html#E2:28_1_2"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="modcat_1.html#E1:28_1_2_2"><i><font color="Green" title="E26">A15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:28_1_2_2"/>
<font color="Maroon" title="c4">c1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="txt" href="modcat_1.html#E4:28_1_2"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="modcat_1.html#E6:28_1_2"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="modcat_1.html#E1:28_1_2_2"><i><font color="Green" title="E26">A15</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:28_1_2"/>
<font color="Maroon" title="c4">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c2</font>
 <b>by </b><i><a class="txt" href="modcat_1.html#E6:28_1_2"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="binop_1.html#T32" title="BINOP_1:th.32">BINOP_1:32</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
