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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>, <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="E1:52"/>
 <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="E2:52"/><b>then </b><i><font color="Green">E38</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="setwiseo.html#K5">\/</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="setwiseo.html#K5">\/</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/>
<a NAME="E3:52"/><i><font color="Green">E39</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="setwiseo.html#K5">\/</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="setwiseo.html#K5">\/</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="normform.html#T67" target="_self">Th67</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_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="E1:52_1"/><i><font color="Green">E40</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="setwiseo.html#K5">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><a NAME="E2:52_1"/><b>then </b><i><font color="Green">E41</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="setwiseo.html#K5">\/</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="52_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><b>that </b><br/><a NAME="E1:52_1_1"/><i><font color="Green">E42</font></i>: 
<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="setwiseo.html#K5">\/</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:52_1_1"/><i><font color="Green">E43</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_1_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> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="normform.html#E1:52_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_1_1_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>
 ;<br/><div class="add"><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="E3:52_1_1_1_1_1"/><i><font color="Green">E44</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="E4:52_1_1_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K9" target="_self">mi</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="normform.html#T65" target="_self">Th65</a></i>;<br/><a NAME="E5:52_1_1_1_1_1"/><i><font color="Green">E46</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="setwiseo.html#K5">\/</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="normform.html#E4:52_1_1_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E6:52_1_1_1_1_1"/>
<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#E2:52_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="normform.html#E3:52_1_1_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="normform.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E7:52_1_1_1_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#E1:52_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="normform.html#E5:52_1_1_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="normform.html#T58" target="_self">Th58</a></i>;<br/><b>hence </b><a NAME="E8:52_1_1_1_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#E2:52_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="normform.html#E3:52_1_1_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="normform.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:52_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</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="setwiseo.html#K5">\/</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E3:52_1_1_1_1_2"/>
<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#E1:52_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="normform.html#E2:52_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="normform.html#T58" target="_self">Th58</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:52_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:52_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="setwiseo.html#K5">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="normform.html#E2:52"><i><font color="Green">E38</font></i></a>, <a class="txt" href="normform.html#E2:52_1"><i><font color="Green">E41</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="E5:52"/>
 <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="setwiseo.html#K5">\/</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="setwiseo.html#K5">\/</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="52_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="E1:52_2"/><i><font color="Green">E40</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="setwiseo.html#K5">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><a NAME="E2:52_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="setwiseo.html#K5">\/</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#E2:52"><i><font color="Green">E38</font></i></a>, <a class="ref" href="normform.html#T58" target="_self">Th58</a></i>;<br/><b>hence </b><a NAME="E3:52_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="setwiseo.html#K5">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="normform.html#E3:52"><i><font color="Green">E39</font></i></a>, <a class="txt" href="normform.html#E1:52_2"><i><font color="Green">E40</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="E7:52"/>
 <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="setwiseo.html#K5">\/</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="setwiseo.html#K5">\/</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/>


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