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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:49"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></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">c<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">c<sub>1</sub></font>] means <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">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 
;<br/>
<a NAME="E3:49"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
<b>by </b><i><a class="ref" href="normform.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E4:49"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>] &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] ) )
 
<b>from </b><i><a class="ref" href="fraenkel.html#S23">FRAENKEL:sch 23</a>(<a class="txt" href="normform.html#E2:49"><i><font color="Green">E36</font></i></a>, <a class="txt" href="normform.html#E3:49"><i><font color="Green">E37</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:49"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E7:49"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E8:49"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="normform.html#R1" target="_self">c=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="normform.html#E1:49"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E9:49"/>
<font color="Maroon">c<sub>4</sub></font> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="normform.html#E7:49"><i><font color="Green">E39</font></i></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">c<sub>5</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:49_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:49_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E3:49_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="normform.html#E8:49"><i><font color="Green">E40</font></i></a>, <a class="txt" href="normform.html#E1:49_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="normform.html#E2:49_1"><i><font color="Green">E42</font></i></a></i>;<br/><b>hence </b><a NAME="E4:49_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="normform.html#E2:49_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="normform.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:49"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K9" target="_self">mi</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="normform.html#E6:49"><i><font color="Green">E38</font></i></a>, <a class="ref" href="normform.html#T61" target="_self">Th61</a></i>;<br/>


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