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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>, <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/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><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#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font>;<br/><i><font color="Green">E40</font></i>:  <a href="heyting1.html#K3" target="_self">@</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K9" target="_self">StrongImpl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="heyting1.html#K9" target="_self">StrongImpl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>

<br/>.= 
 <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">c<sub>2</sub></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">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="heyting1.html#D9" target="_self">Def9</a></i>
;<br/><i><font color="Green">E41</font></i>: <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K9" target="_self">StrongImpl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K9" target="_self">StrongImpl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
 <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">c<sub>2</sub></font></span>)</span> <a href="normform.html#K10">^</a> <span class="p2">(<span class="default"><a href="normform.html#K9">mi</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="heyting1.html#K7" target="_self">=&gt;&gt;</a> <span class="p4">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="heyting1.html#E1:41_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="heyting1.html#E8"><i><font color="Green">Lemma5</font></i></a></i>
<br/>.= 
 <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">c<sub>2</sub></font></span>)</span> <a href="normform.html#K10">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="heyting1.html#K7" target="_self">=&gt;&gt;</a> <span class="p3">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="normform.html#T75">NORMFORM:75</a></i>
;<br/><b>assume </b><a NAME="E3:41_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K9" target="_self">StrongImpl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><a NAME="E4:41_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font></span>)</span> <a href="normform.html#K10">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="heyting1.html#E2:41_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="normform.html#T58">NORMFORM:58</a></i>;<br/><b>hence </b><a NAME="E5:41_1"/>
ex <font color="Olive">b<sub>1</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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="normform.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="heyting1.html#E40"><i><font color="Green">Lemma30</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:41"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K9" target="_self">StrongImpl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="heyting1.html#E24"><i><font color="Green">Lemma16</font></i></a></i>;<br/>


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