<?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> be    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K8">Normal_forms_on</a> <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:35"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10">^</a> <span class="p1">(<span class="default"><a href="heyting1.html#K6" target="_self">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10">^</a> <span class="p1">(<span class="default"><a href="heyting1.html#K6" target="_self">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="heyting1.html#E1:35"><i><font color="Green">E32</font></i></a>, <a class="ref" href="setwiseo.html#T14">SETWISEO:14</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:35"/><i><font color="Green">E33</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>
 <b>and </b><br/><a NAME="E6:35"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="heyting1.html#K6" target="_self">-</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E7:35"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="normform.html#K1">\/</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="heyting1.html#E1:35"><i><font color="Green">E32</font></i></a>, <a class="ref" href="normform.html#T55">NORMFORM:55</a></i>;<br/>
<a NAME="E8:35"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="heyting1.html#E7:35"><i><font color="Green">E35</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1"><a href="heyting1.html#K1" target="_self">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="heyting1.html#K1" target="_self">]</a></span><b> such that </b><br/><a NAME="E10:35"/><i><font color="Green">E37</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">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>2</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span>
 <b>and </b><br/><a NAME="E11:35"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> ,<span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> <a href="domain_1.html#K2">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> </span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="heyting1.html#E6:35"><i><font color="Green">E34</font></i></a>, <a class="ref" href="heyting1.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E12:35"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="heyting1.html#E5:35"><i><font color="Green">E33</font></i></a>, <a class="txt" href="heyting1.html#E10:35"><i><font color="Green">E37</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="domain_1.html#K2">`1</a>  or <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="domain_1.html#K3">`2</a>  )
 </span><b>by </b><i><a class="txt" href="heyting1.html#E12:35"><i><font color="Green">E39</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="35_1_1_1"><b>case </b></a><a NAME="E1:35_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="domain_1.html#K2">`1</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:35_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K2">`1</a> 
 <b>by </b><i><a class="txt" href="heyting1.html#E8:35"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:35_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> <a href="domain_1.html#K2">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="heyting1.html#E5:35"><i><font color="Green">E33</font></i></a>, <a class="txt" href="heyting1.html#E1:35_1_1_1"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E4:35_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="domain_1.html#K3">`2</a> 
 <b>by </b><i><a class="txt" href="heyting1.html#E11:35"><i><font color="Green">E38</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><b>hence </b><a NAME="E5:35_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K3">`2</a> 
 <b>by </b><i><a class="txt" href="heyting1.html#E8:35"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2"><b>case </b></a><a NAME="E1:35_1_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="domain_1.html#K3">`2</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:35_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K3">`2</a> 
 <b>by </b><i><a class="txt" href="heyting1.html#E8:35"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:35_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="heyting1.html#E5:35"><i><font color="Green">E33</font></i></a>, <a class="txt" href="heyting1.html#E1:35_1_1_2"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E4:35_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="domain_1.html#K2">`1</a> 
 <b>by </b><i><a class="txt" href="heyting1.html#E11:35"><i><font color="Green">E38</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><b>hence </b><a NAME="E5:35_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K2">`1</a> 
 <b>by </b><i><a class="txt" href="heyting1.html#E8:35"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E14:35"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E15:35"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K2">`1</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K3">`2</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E16:35"/>
contradiction
 <b>by </b><i><a class="ref" href="normform.html#T44">NORMFORM:44</a></i>;<br/>


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