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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">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="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><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#K8">Normal_forms_on</a> <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><a NAME="E1:34"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="normform.html#K10">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="heyting1.html#K7" target="_self">=&gt;&gt;</a> <span class="p2">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>then 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="E3:34"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:34"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="heyting1.html#K7" target="_self">=&gt;&gt;</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E5:34"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</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="ref" href="normform.html#T55">NORMFORM:55</a></i>;<br/>
<a NAME="E6:34"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">c<sub>3</sub></font>
 
;<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="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E8:34"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E9:34"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="normform.html#K6">FinPairUnion</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K5" target="_self">pair_diff</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcop_1.html#K6">.:</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><a href="funct_3.html#K6">incl</a> <span class="p3">(<span class="default"><a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="heyting1.html#E4:34"><i><font color="Green">E32</font></i></a>, <a class="ref" href="heyting1.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E10:34"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K5" target="_self">pair_diff</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcop_1.html#K6">.:</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><a href="funct_3.html#K6">incl</a> <span class="p3">(<span class="default"><a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="normform.html#K3">\</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="heyting1.html#E38"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<a NAME="E11:34"/><i><font color="Green">E38</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>7</sub></font> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="heyting1.html#E3:34"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</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#E6:34"><i><font color="Green">E34</font></i></a>, <a class="txt" href="heyting1.html#E8:34"><i><font color="Green">E35</font></i></a>, <a class="ref" href="setwiseo.html#T14">SETWISEO:14</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E13:34"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="heyting1.html#E8:34"><i><font color="Green">E35</font></i></a>, <a class="txt" href="heyting1.html#E11:34"><i><font color="Green">E38</font></i></a></i>;<br/>

<a NAME="E14:34"/>
<font color="Maroon">c<sub>8</sub></font> <a href="normform.html#K3">\</a> <font color="Maroon">c<sub>5</sub></font> <a href="normform.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="heyting1.html#E3:34"><i><font color="Green">E31</font></i></a>, <a class="txt" href="heyting1.html#E9:34"><i><font color="Green">E36</font></i></a>, <a class="txt" href="heyting1.html#E10:34"><i><font color="Green">E37</font></i></a>, <a class="ref" href="normform.html#T35">NORMFORM:35</a></i>;<br/>
<b>hence </b><a NAME="E15:34"/>
<font color="Maroon">c<sub>8</sub></font> <a href="normform.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="heyting1.html#E5:34"><i><font color="Green">E33</font></i></a>, <a class="ref" href="normform.html#T30">NORMFORM:30</a></i>;<br/>


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