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<b>let </b><font color="Maroon" title="c1">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="normform.html#K12" title="NORMFORM:func.12">NormForm</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b3">K</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K8" title="NORMFORM:func.8">Normal_forms_on</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">K</font> <a href="heyting1.html#K7" title="HEYTING1:func.7">=&gt;&gt;</a> <span class="p2">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Olive" title="b2">u</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b4">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">u</font> &amp; <font color="Olive" title="b4">b</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Olive" title="b1">a</font> )</span><br/><b>let </b><font color="Maroon" title="c2">a</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="normform.html#K12" title="NORMFORM:func.12">NormForm</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">K</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K8" title="NORMFORM:func.8">Normal_forms_on</a> <font color="Maroon" title="c1">A</font>  st <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">K</font> <a href="heyting1.html#K7" title="HEYTING1:func.7">=&gt;&gt;</a> <span class="p2">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Olive" title="b1">u</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b3">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b3">b</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c2">a</font> )</span><br/><b>let </b><font color="Maroon" title="c3">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="normform.html#K12" title="NORMFORM:func.12">NormForm</a> <font color="Maroon" title="c1">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">K</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K8" title="NORMFORM:func.8">Normal_forms_on</a> <font color="Maroon" title="c1">A</font>  st <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">K</font> <a href="heyting1.html#K7" title="HEYTING1:func.7">=&gt;&gt;</a> <span class="p2">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font> &amp; <font color="Olive" title="b2">b</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c2">a</font> )</span><br/><b>let </b><font color="Maroon" title="c4">K</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K8" title="NORMFORM:func.8">Normal_forms_on</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="heyting1.html#K7" title="HEYTING1:func.7">=&gt;&gt;</a> <span class="p2">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span> implies  ex <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font> &amp; <font color="Olive" title="b1">b</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c2">a</font> ) )</span><br/>







<b>assume </b><a NAME="E1:33"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="heyting1.html#K7" title="HEYTING1:func.7">=&gt;&gt;</a> <span class="p2">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font> &amp; <font color="Olive" title="b1">b</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E3:33"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font>
 <b>and </b><br/><a NAME="E4:33"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> <a href="heyting1.html#K7" title="HEYTING1:func.7">=&gt;&gt;</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">u</font></span>)</span>
 <b>and </b><br/><a NAME="E5:33"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="ref" href="normform.html#T55" title="NORMFORM:th.55">NORMFORM:55</a></i>;<br/>
<a NAME="E6:33"/><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">u</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c7">f</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><b> such that </b><br/><a NAME="E8:33"/><i><font color="Green" title="E32">A5</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c4">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">u</font>
 <b>and </b><br/><a NAME="E9:33"/><i><font color="Green" title="E33">A6</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normform.html#K6" title="NORMFORM:func.6">FinPairUnion</a> <font color="Maroon" title="c4">K</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K5" title="HEYTING1:func.5">pair_diff</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Maroon" title="c7">f</font>,<span class="p2">(<span class="default"><a href="funct_3.html#K6" title="FUNCT_3:func.6">incl</a> <span class="p3">(<span class="default"><a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:33"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="heyting1.html#T23" target="_self" title="HEYTING1:th.23">Th23</a></i>;<br/>
<a NAME="E10:33"/><i><font color="Green" title="E34">A7</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K5" title="HEYTING1:func.5">pair_diff</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Maroon" title="c7">f</font>,<span class="p2">(<span class="default"><a href="funct_3.html#K6" title="FUNCT_3:func.6">incl</a> <span class="p3">(<span class="default"><a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="normform.html#K3" title="NORMFORM:func.3">\</a> <font color="Maroon" title="c5">b</font>
 
<b>by </b><i><a class="txt" href="heyting1.html#E36"><i><font color="Green" title="E25">Lm5</font></i></a></i>;<br/>
<a NAME="E11:33"/><i><font color="Green" title="E35">A8</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c4">K</font>
 
<b>by </b><i><a class="txt" href="#E3:33"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">d</font> = <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="#E6:33"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="#E8:33"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="setwiseo.html#T14" title="SETWISEO:th.14">SETWISEO:14</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">d</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font> &amp; <font color="Maroon" title="c8">d</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<b>thus </b><a NAME="E13:33"/>
<font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font>
 <b>by </b><i><a class="txt" href="#E8:33"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="#E11:33"><i><font color="Green" title="E35">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">d</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c2">a</font></span><br/>

<b>thus </b><a NAME="E14:33"/>
<font color="Maroon" title="c8">d</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="#E3:33"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="#E5:33"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="#E9:33"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="#E10:33"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="normform.html#T30" title="NORMFORM:th.30">NORMFORM:30</a>, <a class="ref" href="normform.html#T35" title="NORMFORM:th.35">NORMFORM:35</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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