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<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">u</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">A</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:42"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font></span>)</span> <a href="normform.html#K10">^</a> <span class="p1"><a href="heyting1.html#K2" target="_self">{</a><span class="default"><font color="Maroon">a</font></span><a href="heyting1.html#K2" target="_self">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</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">c</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>;<br/><b>assume </b><a NAME="E1:42_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="heyting1.html#K4" target="_self">Atom</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E2:42_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:42_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="heyting1.html#K6" target="_self">-</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font></span>)</span>
 <b>and </b><br/><a NAME="E5:42_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">b</font> <a href="normform.html#R1">c=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="txt" href="heyting1.html#E8"><i><font color="Green">Lemma41</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">d</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E7:42_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">d</font> <a href="normform.html#R1">c=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E8:42_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K9">mi</a> <span class="p1">(<span class="default"><a href="heyting1.html#K6" target="_self">-</a> <span class="p2">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="heyting1.html#E9"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="normform.html#T65">NORMFORM:65</a></i>;<br/><b>take </b><font color="Maroon">e</font> = <font color="Maroon">d</font>;<br/><b>thus </b><a NAME="E9:42_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="heyting1.html#K8" target="_self">pseudo_compl</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="heyting1.html#T7" target="_self">Th7</a>, </i>;<br/><b>thus </b><a NAME="E10:42_1"/>
<font color="Maroon">e</font> <a href="normform.html#R1">c=</a> <font color="Maroon">c</font>
 <b>by </b><i>, , <a class="ref" href="normform.html#T5">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" target="_self">Atom</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><a href="heyting1.html#K8" target="_self">pseudo_compl</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>
 <b>by </b><i/>;<br/>


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