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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>, <font color="Olive" title="b3">w</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>  st ( for <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 <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> holds <br/><font color="Olive" title="b4">b</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> ) &amp; <font color="Olive" title="b2">u</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<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="Olive" title="b1">a</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b3">w</font> holds <br/><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="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><a href="heyting1.html#K9" title="HEYTING1:func.9">StrongImpl</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b2">u</font>,<font color="Olive" title="b3">w</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>, <font color="Olive" title="b2">w</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>  st ( for <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 <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> holds <br/><font color="Olive" title="b3">b</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> ) &amp; <font color="Olive" title="b1">u</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">w</font> holds <br/><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#K9" title="HEYTING1:func.9">StrongImpl</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">w</font></span><br/><b>let </b><font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">w</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">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 <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> holds <br/><font color="Olive" title="b1">b</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> ) &amp; <font color="Maroon" title="c3">u</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">w</font> implies <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#K9" title="HEYTING1:func.9">StrongImpl</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">w</font> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:42"/><i><font color="Green" title="E39">A1</font></i>: 
for <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 <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> holds <br/><font color="Olive" title="b1">b</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font>
 <b>and </b><br/><a NAME="E2:42"/><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon" title="c3">u</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">w</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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#K9" title="HEYTING1:func.9">StrongImpl</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">w</font></span><br/>

<div><i><font color="Green" title="E41">A3</font></i>: <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="c5">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font> implies  ex <font color="Olive" title="b1">e</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">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Olive" title="b1">e</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">c</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c2">a</font> ) )</span><br/><b>assume </b><a NAME="E1:42_1"/><i><font color="Green" title="E42">A4</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font>
 ; <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">e</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">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Olive" title="b1">e</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">c</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c2">a</font> )</span><br/><a NAME="E2:42_1"/><i><font color="Green" title="E43">A5</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>
 <b>by </b><i><a class="ref" href="heyting1.html#T12" target="_self" title="HEYTING1:th.12">Th12</a></i>;<br/><a NAME="E3:42_1"/><i><font color="Green" title="E44">A6</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c2">a</font> is    <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="#E1:42"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E1:42_1"><i><font color="Green" title="E42">A4</font></i></a></i>;<br/><a NAME="E4:42_1"/>
<font color="Maroon" title="c5">c</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <span class="p2">(<span class="default"><span class="p3">(<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></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:42"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E1:42_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="#E2:42_1"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="normform.html#T56" title="NORMFORM:th.56">NORMFORM:56</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">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="E6:42_1"/><i><font color="Green" title="E45">A7</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">c</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c2">a</font>
 <b>and </b><br/><a NAME="E7:42_1"/><i><font color="Green" title="E46">A8</font></i>: 
<font color="Maroon" title="c6">b</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"><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> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <span class="p2">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <span class="p3">(<span class="default"><span class="p4">(<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></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:42_1"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="normform.html#T65" title="NORMFORM:th.65">NORMFORM:65</a></i>;<br/><a NAME="E8:42_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">A</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">u</font>,<span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>
 <b>by </b><i><a class="txt" href="#E7:42_1"><i><font color="Green" title="E46">A8</font></i></a>, <a class="ref" href="normform.html#D14" title="NORMFORM:def.14">NORMFORM:def 14</a></i>;<br/><a NAME="E9:42_1"/><b>then </b>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>
 <b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">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="E11:42_1"/><i><font color="Green" title="E47">A9</font></i>: 
<font color="Maroon" title="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">w</font>
 <b>and </b><br/><a NAME="E12:42_1"/><i><font color="Green" title="E48">A10</font></i>: 
<font color="Maroon" title="c7">d</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="#E2:42"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="heyting1.html#E21"><i><font color="Green" title="E12">Lm2</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c8">e</font> = <font color="Maroon" title="c7">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">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Maroon" title="c8">e</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">c</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c2">a</font> )</span><br/><b>thus </b><a NAME="E13:42_1"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">w</font>
 <b>by </b><i><a class="txt" href="#E11:42_1"><i><font color="Green" title="E47">A9</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">e</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">c</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c2">a</font></span><br/><b>thus </b><a NAME="E14:42_1"/>
<font color="Maroon" title="c8">e</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">c</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="#E6:42_1"><i><font color="Green" title="E45">A7</font></i></a>, <a class="txt" href="#E12:42_1"><i><font color="Green" title="E48">A10</font></i></a>, <a class="ref" href="normform.html#T5" title="NORMFORM:th.5">NORMFORM:5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">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> implies  ex <font color="Olive" title="b1">e</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">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="heyting1.html#K9" title="HEYTING1:func.9">StrongImpl</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">w</font> &amp; <font color="Olive" title="b1">e</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">c</font> ) )</span><br/><b>assume </b><a NAME="E1:42_2"/>
<font color="Maroon" title="c5">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>
 ; <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">e</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">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="heyting1.html#K9" title="HEYTING1:func.9">StrongImpl</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">w</font> &amp; <font color="Olive" title="b1">e</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">c</font> )</span><br/><a NAME="E2:42_2"/><b>then </b>
<font color="Maroon" title="c5">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="c6">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_2"/><i><font color="Green" title="E42">A11</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <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="c4">w</font></span>)</span>
 <b>and </b><br/><a NAME="E5:42_2"/><i><font color="Green" title="E43">A12</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">c</font>
 <b>by </b><i><a class="txt" href="#E1:42"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E3:42"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="heyting1.html#T25" target="_self" title="HEYTING1:th.25">Th25</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">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_2"/><i><font color="Green" title="E44">A13</font></i>: 
<font color="Maroon" title="c7">d</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c6">b</font>
 <b>and </b><br/><a NAME="E8:42_2"/><i><font color="Green" title="E45">A14</font></i>: 
<font color="Maroon" title="c7">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"><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> <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="c4">w</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:42_2"><i><font color="Green" title="E42">A11</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="c8">e</font> = <font color="Maroon" title="c7">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">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="heyting1.html#K9" title="HEYTING1:func.9">StrongImpl</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">w</font> &amp; <font color="Maroon" title="c8">e</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">c</font> )</span><br/><b>thus </b><a NAME="E9:42_2"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="heyting1.html#K9" title="HEYTING1:func.9">StrongImpl</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">w</font>
 <b>by </b><i><a class="txt" href="#E8:42_2"><i><font color="Green" title="E45">A14</font></i></a>, <a class="ref" href="heyting1.html#D9" target="_self" title="HEYTING1:def.9">Def9</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">e</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">c</font></span><br/><b>thus </b><a NAME="E10:42_2"/>
<font color="Maroon" title="c8">e</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">c</font>
 <b>by </b><i><a class="txt" href="#E5:42_2"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="#E7:42_2"><i><font color="Green" title="E44">A13</font></i></a>, <a class="ref" href="normform.html#T5" title="NORMFORM:th.5">NORMFORM:5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:42"/>
<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#K9" title="HEYTING1:func.9">StrongImpl</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">w</font>
 <b>by </b><i><a class="txt" href="heyting1.html#E22"><i><font color="Green" title="E13">Lm3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
