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<b>let </b><font color="Maroon">n1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:35_1_2"/><i><font color="Green">E35</font></i>: 
 ex <font color="Olive">X</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> iff ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font> &amp; the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) ) ) &amp; <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Olive">X</font> )
 <b>and </b><br/><a NAME="E2:35_1_2"/><i><font color="Green">E36</font></i>: 
 ex <font color="Olive">X</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> iff ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font> &amp; the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) ) ) &amp; <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Olive">X</font> )
 ;<br/>

<b>consider </b><font color="Maroon">X1</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:35_1_2"/><i><font color="Green">E37</font></i>: 
( ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> iff ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font> &amp; the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) ) ) &amp; <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X1</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">X2</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:35_1_2"/><i><font color="Green">E38</font></i>: 
( ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> iff ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font> &amp; the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) ) ) &amp; <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X2</font> )
 <b>by </b><i/>;<br/>
<a NAME="E7:35_1_2"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> iff <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:35_1_2_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> iff ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font> &amp; the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:35_1_2_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> iff <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> )
 <b>by </b><i><a class="ref" href="graph_1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:35_1_2"/>
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 <b>by </b><i>, <a class="ref" href="graph_1.html#D1" target="_self">Def1</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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