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





<b>assume </b><a NAME="E1:49"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font>,   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E2:49"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">a</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  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">a</font>]
 
;<br/>
<a NAME="E3:49"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</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 <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>,<font color="Olive">c</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">c</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E4:49"/>
for <font color="Olive">a</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 <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/> ex <font color="Olive">b</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="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>,<font color="Olive">a</font>] &amp; ( for <font color="Olive">c</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 <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>,<font color="Olive">b</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>,<font color="Olive">c</font>] ) )
 
<b>from </b><i><a class="ref" href="fraenkel.html#S23">FRAENKEL:sch 23</a>(<a class="txt" href="normform.html#E39"><i><font color="Green">Lemma36</font></i></a>, <a class="txt" href="normform.html#E40"><i><font color="Green">Lemma37</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">c</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><b> such that </b><br/><a NAME="E6:49"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E7:49"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E8:49"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">a</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 <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">a</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">c</font> holds <br/><font color="Maroon">c</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Olive">a</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">c</font>
;<br/>

<b>thus </b><a NAME="E9:49"/>
<font color="Maroon">c</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="normform.html#T36" target="_self">Th36</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</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><b>that </b><br/><a NAME="E1:49_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:49_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">b</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">c</font>
 ;<br/><a NAME="E3:49_1"/>
<font color="Maroon">c</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="normform.html#T37" target="_self">Th37</a>, <a class="ref" href="normform.html#T38" target="_self">Th38</a>, <a class="ref" href="normform.html#T39" target="_self">Th39</a></i>;<br/><b>hence </b><a NAME="E4:49_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="normform.html#T39" target="_self">Th39</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:49"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K9" target="_self">mi</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="normform.html#E41"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="normform.html#D6" target="_self">Def6</a></i>;<br/>


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