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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>) <b>-&gt; </b>   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of $1,$1 =  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> $1;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>] <b>means </b>verum;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>] <b>means </b><font color="Maroon">H<sub>1</sub></font>($1) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">C</font>;<br/>
<b>set </b><font color="Maroon" title="c2">M</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) where <font color="Olive" title="b1">a</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c3">N</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) where <font color="Olive" title="b1">a</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> ;<br/>
<a NAME="E1:74_1_1"/><i><font color="Green" title="E56">A1</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] )
 
;<br/>
<a NAME="E2:74_1_1"/><i><font color="Green" title="E57">A2</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) where <font color="Olive" title="b1">a</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) where <font color="Olive" title="b1">a</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S3" title="FRAENKEL:sch.3">FRAENKEL:sch 3</a>(<a class="txt" href="nattra_1.html#E1:74_1_1"><i><font color="Green" title="E56">A1</font></i></a>);<br/></i>
<b>set </b><font color="Maroon" title="c4">N'</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) where <font color="Olive" title="b1">a</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] ) </span>}</span> ;<br/>
<a NAME="E3:74_1_1"/><i><font color="Green" title="E58">A3</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) where <font color="Olive" title="b1">a</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">C</font>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S17" title="FRAENKEL:sch.17">FRAENKEL:sch 17</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>] <b>means </b>( <font color="Maroon">H<sub>1</sub></font>($1) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon">S<sub>1</sub></font>[$1] );<br/>
<b>set </b><font color="Maroon" title="c5">N''</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) where <font color="Olive" title="b1">a</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> ;<br/>
<a NAME="E4:74_1_1"/><i><font color="Green" title="E59">A4</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] iff <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">a</font>] )
 
;<br/>
<a NAME="E5:74_1_1"/><i><font color="Green" title="E60">A5</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) where <font color="Olive" title="b1">a</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) where <font color="Olive" title="b1">a</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S3" title="FRAENKEL:sch.3">FRAENKEL:sch 3</a>(<a class="txt" href="nattra_1.html#E4:74_1_1"><i><font color="Green" title="E59">A4</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c6">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>;<br/>
<a NAME="E7:74_1_1"/>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) where <font color="Olive" title="b1">a</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">M</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) where <font color="Olive" title="b1">a</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="txt" href="nattra_1.html#E2:74_1_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="nattra_1.html#E3:74_1_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="txt" href="nattra_1.html#E5:74_1_1"><i><font color="Green" title="E60">A5</font></i></a></i>;<br/>
<a NAME="E9:74_1_1"/><i><font color="Green" title="E61">A6</font></i>: 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">M</font> is   <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"><font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c7">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#T43" title="PARTFUN1:th.43">PARTFUN1:43</a></i>;<br/>
<a NAME="E10:74_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">M</font></span>)</span> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font> )</span><br/>

<b>assume </b><a NAME="E1:74_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">M</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font></span><br/>

<b>then consider </b><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:74_1_1_1"/><i><font color="Green" title="E62">A7</font></i>: 
<font color="Maroon" title="c9">y</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> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">M</font></span>)</span>
 <b>and </b><br/><a NAME="E4:74_1_1_1"/><i><font color="Green" title="E63">A8</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:74_1_1_1"/><i><font color="Green" title="E64">A9</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c7">M</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:74_1_1_1"><i><font color="Green" title="E62">A7</font></i></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E6:74_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c7">M</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">y1</font>, <font color="Maroon" title="c11">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:74_1_1_1"/><i><font color="Green" title="E65">A10</font></i>: 
( <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Maroon" title="c11">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font> )
 <b>and </b><br/><a NAME="E9:74_1_1_1"/><i><font color="Green" title="E66">A11</font></i>: 
<font color="Maroon" title="c9">y</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="c10">y1</font>,<font color="Maroon" title="c11">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">a1</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E11:74_1_1_1"/><i><font color="Green" title="E67">A12</font></i>: 
<font color="Maroon" title="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">a1</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E8:74_1_1_1"><i><font color="Green" title="E65">A10</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">a2</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E13:74_1_1_1"/><i><font color="Green" title="E68">A13</font></i>: 
<font color="Maroon" title="c11">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c13">a2</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E8:74_1_1_1"><i><font color="Green" title="E65">A10</font></i></a></i>;<br/>
<a NAME="E14:74_1_1_1"/><i><font color="Green" title="E69">A14</font></i>: 
<font color="Maroon" title="c9">y</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> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E5:74_1_1_1"><i><font color="Green" title="E64">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E15:74_1_1_1"/><i><font color="Green" title="E70">A15</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c12">a1</font>,<font color="Maroon" title="c12">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_1.html#T56" title="CAT_1:th.56">CAT_1:56</a></i>;<br/>
<i><font color="Green" title="E71">A16</font></i>: <font color="Maroon" title="c12">a1</font> = 
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">a1</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
<br/>.= 
 <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c13">a2</font></span>)</span>
<b>by </b><i><a class="txt" href="nattra_1.html#E9:74_1_1_1"><i><font color="Green" title="E66">A11</font></i></a>, <a class="txt" href="nattra_1.html#E11:74_1_1_1"><i><font color="Green" title="E67">A12</font></i></a>, <a class="txt" href="nattra_1.html#E13:74_1_1_1"><i><font color="Green" title="E68">A13</font></i></a>, <a class="txt" href="nattra_1.html#E14:74_1_1_1"><i><font color="Green" title="E69">A14</font></i></a>, <a class="ref" href="cat_1.html#T40" title="CAT_1:th.40">CAT_1:40</a></i>
<br/>.= 
<font color="Maroon" title="c13">a2</font>
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
;<br/>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">a1</font> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">a1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">a1</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T59" title="CAT_1:th.59">CAT_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">a1</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c13">a2</font></span>)</span>
<b>by </b><i><a class="txt" href="nattra_1.html#E15:74_1_1_1"><i><font color="Green" title="E70">A15</font></i></a>, <a class="txt" href="nattra_1.html#E16:74_1_1_1"><i><font color="Green" title="E71">A16</font></i></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></i>
<br/>.= 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">a1</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c13">a2</font></span>)</span>
<b>by </b><i><a class="txt" href="nattra_1.html#E9:74_1_1_1"><i><font color="Green" title="E66">A11</font></i></a>, <a class="txt" href="nattra_1.html#E11:74_1_1_1"><i><font color="Green" title="E67">A12</font></i></a>, <a class="txt" href="nattra_1.html#E13:74_1_1_1"><i><font color="Green" title="E68">A13</font></i></a>, <a class="txt" href="nattra_1.html#E14:74_1_1_1"><i><font color="Green" title="E69">A14</font></i></a>, <a class="ref" href="cat_1.html#D4" title="CAT_1:def.4">CAT_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c8">x</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E4:74_1_1_1"><i><font color="Green" title="E63">A8</font></i></a>, <a class="txt" href="nattra_1.html#E5:74_1_1_1"><i><font color="Green" title="E64">A9</font></i></a>, <a class="txt" href="nattra_1.html#E9:74_1_1_1"><i><font color="Green" title="E66">A11</font></i></a>, <a class="txt" href="nattra_1.html#E11:74_1_1_1"><i><font color="Green" title="E67">A12</font></i></a>, <a class="txt" href="nattra_1.html#E13:74_1_1_1"><i><font color="Green" title="E68">A13</font></i></a>, <a class="ref" href="funct_1.html#T71" title="FUNCT_1:th.71">FUNCT_1:71</a></i>
;<br/>
<b>hence </b><a NAME="E18:74_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font>
 ; <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>then reconsider </b><font color="Maroon" title="c8">COMP</font> = the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">M</font> as   <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"><font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c7">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c7">M</font> <b>by </b><i><a class="txt" href="nattra_1.html#E9:74_1_1"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="partfun1.html#T29" title="PARTFUN1:th.29">PARTFUN1:29</a></i>;<br/>
<a NAME="E12:74_1_1"/><i><font color="Green" title="E62">A17</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:74_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font> )</span><br/>

<b>assume </b><a NAME="E1:74_1_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font></span><br/>

<b>then consider </b><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:74_1_1_2"/><i><font color="Green" title="E63">A18</font></i>: 
<font color="Maroon" title="c10">y</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> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E4:74_1_1_2"/><i><font color="Green" title="E64">A19</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">y</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="txt" href="nattra_1.html#E3:74_1_1_2"><i><font color="Green" title="E63">A18</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:74_1_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c11">y</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E4:74_1_1_2"><i><font color="Green" title="E64">A19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:74_1_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font>
 ; <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>then reconsider </b><font color="Maroon" title="c9">ID</font> = the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c7">M</font> <b>by </b><i><a class="txt" href="nattra_1.html#E12:74_1_1"><i><font color="Green" title="E62">A17</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<a NAME="E15:74_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E16:74_1_1"/><b>then </b><i><font color="Green" title="E63">A20</font></i>: 
the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>;<br/>
<a NAME="E17:74_1_1"/>
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">A</font> =  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c7">M</font>,<span class="p1">(<span class="default">the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Maroon" title="c7">M</font></span>)</span>,<span class="p1">(<span class="default">the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Maroon" title="c7">M</font></span>)</span>,<font color="Maroon" title="c8">COMP</font>,<font color="Maroon" title="c9">ID</font> #) as    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="txt" href="nattra_1.html#E16:74_1_1"><i><font color="Green" title="E63">A20</font></i></a>, <a class="ref" href="nattra_1.html#T9" target="_self" title="NATTRA_1:th.9">Th9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c10">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c10">A</font> st <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Olive" title="b1">a</font></span><br/><a NAME="E1:74_1_1_3"/>
<font color="Maroon" title="c11">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c12">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E3:74_1_1_3"/><i><font color="Green" title="E64">A21</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">a</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c12">a</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c10">A</font> ;<br/><b>take </b><font color="Maroon" title="c14">a</font> = <font color="Maroon" title="c13">a</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c14">a</font></span><br/><b>thus </b><a NAME="E5:74_1_1_3"/>
<font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c14">a</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E3:74_1_1_3"><i><font color="Green" title="E64">A21</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>then reconsider </b><font color="Maroon" title="c11">A</font> = <font color="Maroon" title="c10">A</font> as  <a href="cat_1.html#V1" title="CAT_1:attr.1">strict</a> <a href="nattra_1.html#V2" title="NATTRA_1:attr.2">discrete</a>  <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="ref" href="nattra_1.html#D19" target="_self" title="NATTRA_1:def.19">Def19</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">A</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c11">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C</font> &amp; the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c11">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : verum </span>}</span>  )</span><br/>

<b>thus </b><a NAME="E21:74_1_1"/>
( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c11">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C</font> &amp; the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c11">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : verum </span>}</span>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
