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<b>let </b><font color="Maroon">IT1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">IT2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:37_1_2"/>
(  ex <font color="Olive">X1</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">X1</font> iff ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Olive">z</font> ) ) ) &amp; <font color="Maroon">IT1</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Olive">X1</font> ) &amp;  ex <font color="Olive">X2</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">X2</font> iff ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Olive">z</font> ) ) ) &amp; <font color="Maroon">IT2</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Olive">X2</font> ) )
 ;<br/>

<b>then consider </b><font color="Maroon">X1</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> , <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="E3:37_1_2"/><i><font color="Green">E25</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="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Olive">z</font> ) ) ) &amp; <font color="Maroon">IT1</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X1</font> &amp; ( 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="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Olive">z</font> ) ) ) &amp; <font color="Maroon">IT2</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X2</font> )
 ;<br/>
<a NAME="E4:37_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 ;<br/>

<a NAME="E2:37_1_2_1"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:37_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:37_1_2"/>
<font color="Maroon">X2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_1_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 ;<br/>

<a NAME="E2:37_1_2_2"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:37_1_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:37_1_2"/>
<font color="Maroon">IT1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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