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<b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> holds <br/> ( ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Olive">t</font> ) iff <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> ) </span>}</span> ;<br/>
<a NAME="E1:41_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:41_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> holds <br/> ( ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Olive">t</font> ) iff <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> ) </span>}</span> 
 ;<br/><a NAME="E2:41_1_1_1"/><b>then </b>
 ex <font color="Olive">t</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> &amp; ( for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> holds <br/> ( ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Olive">t</font> ) iff <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> ) ) )
 ;<br/><b>hence </b><a NAME="E3:41_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E3:41_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> holds <br/> ( ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Olive">t</font> ) iff <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:41_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> holds <br/> ( ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Olive">t</font> ) iff <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="normform.html#T28" target="_self">Th28</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:41_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> holds <br/> ( ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Olive">t</font> ) iff <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="normform.html#E1:41_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">M'</font> = <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> holds <br/> ( ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Olive">t</font> ) iff <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> ) </span>}</span>  as    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="txt" href="normform.html#E3:41_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font>, <font color="Maroon">d</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font>;<br/><b>assume </b><a NAME="E1:41_1_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> holds <br/> ( ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Olive">t</font> ) iff <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> ) </span>}</span> 
 ;<br/><a NAME="E2:41_1_1_2"/><b>then </b>
 ex <font color="Olive">t</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> &amp; ( for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> holds <br/> ( ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Olive">t</font> ) iff <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> ) ) )
 ;<br/><a NAME="E3:41_1_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><b>assume </b><a NAME="E4:41_1_1_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> holds <br/> ( ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Olive">t</font> ) iff <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> ) </span>}</span> 
 ;<br/><a NAME="E5:41_1_1_2"/><b>then </b>
 ex <font color="Olive">t</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> st <br/>( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> &amp; ( for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> holds <br/> ( ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Olive">t</font> ) iff <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> ) ) )
 ;<br/><b>hence </b><a NAME="E6:41_1_1_2"/>
( <font color="Maroon">c</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">d</font> implies <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="normform.html#D6" target="_self">Def6</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:41_1_1"/><b>then </b>
<font color="Maroon">M'</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K8" target="_self">Normal_forms_on</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="normform.html#T26" target="_self">Th26</a></i>;<br/>
<b>hence </b><a NAME="E9:41_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> holds <br/> ( ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Olive">t</font> ) iff <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> ) </span>}</span>  is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K8" target="_self">Normal_forms_on</a> <font color="Maroon">A</font>
 ;<br/>


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