<?xml version="1.0"?>
<div class="add">

<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>, <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/>





<a NAME="E1:58"/>
 <a href="normform.html#K9" target="_self">mi</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="normform.html#T64" target="_self">Th64</a></i>;<br/>
<a NAME="E2:58"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="normform.html#K9" target="_self">mi</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="normform.html#T70" target="_self">Th70</a></i>;<br/>
<a NAME="E3:58"/><i><font color="Green">E45</font></i>: 
 <a href="normform.html#K9" target="_self">mi</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="normform.html#K9" target="_self">mi</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="normform.html#T71" target="_self">Th71</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</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><a NAME="E1:58_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K9" target="_self">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="normform.html#K9" target="_self">mi</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><a NAME="E2:58_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="normform.html#K9" target="_self">mi</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="normform.html#T58" target="_self">Th58</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_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><a NAME="E1:58_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</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="E3:58_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E4:58_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="normform.html#K9" target="_self">mi</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="normform.html#E87"><i><font color="Green">Lemma39</font></i></a></i>;<br/><b>assume </b><a NAME="E5:58_1_1"/><i><font color="Green">E50</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="E6:58_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="normform.html#E3:58_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="normform.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E7:58_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</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#E1:58_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="normform.html#E4:58_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="normform.html#T58" target="_self">Th58</a></i>;<br/><b>hence </b><a NAME="E8:58_1_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#E3:58_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="normform.html#E5:58_1_1"><i><font color="Green">E50</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="E4:58_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K9" target="_self">mi</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="normform.html#E2:58"><i><font color="Green">E44</font></i></a>, <a class="txt" href="normform.html#E2:58_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="normform.html#T61" target="_self">Th61</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:58"/>
 <a href="normform.html#K9" target="_self">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="normform.html#K9" target="_self">mi</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="normform.html#K9" target="_self">mi</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="normform.html#E77"><i><font color="Green">Lemma32</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</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><a NAME="E1:58_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K9" target="_self">mi</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><a NAME="E2:58_2"/><b>then </b>
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> <span class="p1">(<span class="default"><a href="normform.html#K9" target="_self">mi</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normform.html#K10" target="_self">^</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="Olive">b<sub>1</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#E2:58"><i><font color="Green">E44</font></i></a>, <a class="ref" href="normform.html#T58" target="_self">Th58</a></i>;<br/><b>hence </b><a NAME="E3:58_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K9" target="_self">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="normform.html#K9" target="_self">mi</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="normform.html#E3:58"><i><font color="Green">E45</font></i></a>, <a class="txt" href="normform.html#E1:58_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="normform.html#T61" target="_self">Th61</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:58"/>
 <a href="normform.html#K9" target="_self">mi</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="normform.html#K9" target="_self">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="normform.html#K9" target="_self">mi</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="normform.html#E77"><i><font color="Green">Lemma32</font></i></a></i>;<br/>


</div>
