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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>let </b><font color="Maroon">w</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><b>that </b><br/><a NAME="E1:43"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">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>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> holds <br/><font color="Olive">b</font> <a href="normform.html#K1">\/</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E2:43"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">u</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">w</font>
 ;<br/>

<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_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="E1:43_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font>
 ;<br/><a NAME="E2:43_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font>
 ;<br/><a NAME="E3:43_1"/>
<font color="Maroon">a</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>
 <b>by </b><i/>;<br/><a NAME="E4:43_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="heyting1.html#K3" target="_self">@</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>
 ;<br/><a NAME="E5:43_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c</font> <a href="normform.html#K1">\/</a> <font color="Maroon">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="heyting1.html#E9"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="heyting1.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E6:43_1"/><b>then </b>
<font color="Maroon">c</font> <a href="normform.html#K1">\/</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <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">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <span class="p2">(<span class="default"><span class="p3">(<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></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="heyting1.html#T8" target="_self">Th8</a>, , <a class="ref" href="normform.html#T56">NORMFORM:56</a></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="E8:43_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">b</font> <a href="normform.html#R1">c=</a> <font color="Maroon">c</font> <a href="normform.html#K1">\/</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E9:43_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K9">mi</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <span class="p3">(<span class="default"><span class="p4">(<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></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="heyting1.html#T10" target="_self">Th10</a>, <a class="ref" href="normform.html#T65">NORMFORM:65</a></i>;<br/><a NAME="E10:43_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">A</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">u</font>,<span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E11:43_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>
 <b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>;<br/><b>then 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="E13:43_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E14:43_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">d</font> <a href="normform.html#R1">c=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="heyting1.html#T1" target="_self">Th1</a></i>;<br/><b>take </b><font color="Maroon">e</font> = <font color="Maroon">d</font>;<br/><b>thus </b><a NAME="E15:43_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">w</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E16:43_1"/>
<font color="Maroon">e</font> <a href="normform.html#R1">c=</a> <font color="Maroon">c</font> <a href="normform.html#K1">\/</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="heyting1.html#D4" target="_self">Def4</a>, <a class="txt" href="heyting1.html#E3:43"><i><font color="Green">E52</font></i></a>, <a class="ref" href="normform.html#T5">NORMFORM:5</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><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="E1:43_2"/>
<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="E2:43_2"/><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="E4:43_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font></span>)</span> <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">w</font></span>)</span>
 <b>and </b><br/><a NAME="E5:43_2"/><i><font color="Green">E68</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#E9"><i><font color="Green">Lemma42</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="E7:43_2"/><i><font color="Green">E69</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="E8:43_2"/><i><font color="Green">E70</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"><span class="p2">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font></span>)</span> <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">w</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="heyting1.html#T7" target="_self">Th7</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="E9:43_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="heyting1.html#K9" target="_self">StrongImpl</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">u</font>,<font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="heyting1.html#T10" target="_self">Th10</a>, </i>;<br/><b>thus </b><a NAME="E10:43_2"/>
<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="heyting1.html#T8" target="_self">Th8</a>, , <a class="ref" href="normform.html#T5">NORMFORM:5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:43"/>
<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#K9" target="_self">StrongImpl</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">u</font>,<font color="Maroon">w</font>
 <b>by </b><i/>;<br/>


</div>
