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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> ;<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>;<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>;<br/>





<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E39">A1</font></i>: 
<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> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <span class="p1"><a href="heyting1.html#K2" title="HEYTING1:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="heyting1.html#K2" title="HEYTING1:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">c</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>;<br/><b>assume </b><a NAME="E1:42_1"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="heyting1.html#K4" title="HEYTING1:func.4">Atom</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font>
 ;<br/><a NAME="E2:42_1"/><b>then </b>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="heyting1.html#T11" target="_self" title="HEYTING1:th.11">Th11</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">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><b> such that </b><br/><a NAME="E4:42_1"/><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting1.html#K6" title="HEYTING1:func.6">-</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="E5:42_1"/><i><font color="Green" title="E41">A3</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="heyting1.html#E1:42"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="heyting1.html#T24" target="_self" title="HEYTING1:th.24">Th24</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">d</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="E7:42_1"/><i><font color="Green" title="E42">A4</font></i>: 
<font color="Maroon" title="c6">d</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">b</font>
 <b>and </b><br/><a NAME="E8:42_1"/><i><font color="Green" title="E43">A5</font></i>: 
<font color="Maroon" title="c6">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><a href="heyting1.html#K6" title="HEYTING1:func.6">-</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>
 <b>by </b><i><a class="txt" href="heyting1.html#E4:42_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="normform.html#T65" title="NORMFORM:th.65">NORMFORM:65</a></i>;<br/><b>take </b><font color="Maroon" title="c7">e</font> = <font color="Maroon" title="c6">d</font>;<br/><b>thus </b><a NAME="E9:42_1"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="heyting1.html#K8" title="HEYTING1:func.8">pseudo_compl</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">u</font>
 <b>by </b><i><a class="txt" href="heyting1.html#E8:42_1"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="heyting1.html#D8" target="_self" title="HEYTING1:def.8">Def8</a></i>;<br/><b>thus </b><a NAME="E10:42_1"/>
<font color="Maroon" title="c7">e</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="heyting1.html#E5:42_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="heyting1.html#E7:42_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="normform.html#T5" title="NORMFORM:th.5">NORMFORM:5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:42"/>
<span class="p1">(<span class="default"><a href="heyting1.html#K4" title="HEYTING1:func.4">Atom</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><a href="heyting1.html#K8" title="HEYTING1:func.8">pseudo_compl</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">u</font>
 <b>by </b><i><a class="txt" href="heyting1.html#E22"><i><font color="Green" title="E14">Lm3</font></i></a></i>;<br/>


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