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<div class="add">

<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">u</font>, <font color="Maroon" title="c3">v</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="E1:18"/><i><font color="Green" title="E14">A1</font></i>: 
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>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font> holds <br/> ex <font color="Olive" title="b2">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 <br/>( <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Olive" title="b2">b</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Olive" title="b1">a</font> )
 ;<br/>

<a NAME="E2:18"/><i><font color="Green" title="E15">A2</font></i>: 
 <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="c2">u</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="heyting1.html#R1" title="HEYTING1:pred.1">c=</a>  <a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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>; <i><font color="Red">:: according to </font></i><a class="ref" href="heyting1.html#D1" target="_self" title="HEYTING1:def.1">HEYTING1:def 1</a><br/>

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c4">a</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="c2">u</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:18_1"/><b>then </b>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">v</font>
 
<b>by </b><i><a class="ref" href="normform.html#T58" title="NORMFORM:th.58">NORMFORM:58</a></i>;<br/>
<a NAME="E3:18_1"/><b>then </b>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font> or <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</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="E5:18_1"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font>
 <b>and </b><br/><a NAME="E6:18_1"/><i><font color="Green" title="E18">A5</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">a</font>
 <b>by </b><i><a class="txt" href="heyting1.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a></i>;<br/>
<a NAME="E7:18_1"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">v</font>
 
<b>by </b><i><a class="txt" href="heyting1.html#E5:18_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E8:18_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">v</font>
 <b>by </b><i><a class="txt" href="heyting1.html#E1:18_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="heyting1.html#E5:18_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="heyting1.html#E6:18_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="normform.html#T60" title="NORMFORM:th.60">NORMFORM:60</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:18"/><i><font color="Green" title="E16">A6</font></i>: 
 <a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">v</font> <a href="heyting1.html#R1" title="HEYTING1:pred.1">c=</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="c2">u</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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>; <i><font color="Red">:: according to </font></i><a class="ref" href="heyting1.html#D1" target="_self" title="HEYTING1:def.1">HEYTING1:def 1</a><br/>

<b>assume </b><a NAME="E1:18_2"/><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">v</font>
 ;<br/>

<a NAME="E2:18_2"/><b>then </b><i><font color="Green" title="E18">A8</font></i>: 
<font color="Maroon" title="c4">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="c2">u</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">v</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:18_2"/><i><font color="Green" title="E19">A9</font></i>: 
<font color="Maroon" title="c4">a</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#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">v</font></span>)</span>
 
<b>by </b><i><a class="txt" href="heyting1.html#E1:18_2"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="normform.html#T66" title="NORMFORM:th.66">NORMFORM:66</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">b</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><b>that </b><br/><a NAME="E1:18_2_1"/><i><font color="Green" title="E20">A10</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">v</font>
 <b>and </b><br/><a NAME="E2:18_2_1"/><i><font color="Green" title="E21">A11</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">a</font>
 ;<br/><a NAME="E3:18_2_1"/>
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font> or <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> )
 <b>by </b><i><a class="txt" href="heyting1.html#E1:18_2_1"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">c</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="E5:18_2_1"/><i><font color="Green" title="E22">A12</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font>
 <b>and </b><br/><a NAME="E6:18_2_1"/><i><font color="Green" title="E23">A13</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="txt" href="heyting1.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a></i>;<br/><a NAME="E7:18_2_1"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="heyting1.html#E3:18_2"><i><font color="Green" title="E19">A9</font></i></a>, <a class="txt" href="heyting1.html#E2:18_2_1"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="heyting1.html#E5:18_2_1"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="heyting1.html#E6:18_2_1"><i><font color="Green" title="E23">A13</font></i></a>, <a class="ref" href="normform.html#T5" title="NORMFORM:th.5">NORMFORM:5</a>, <a class="ref" href="normform.html#T60" title="NORMFORM:th.60">NORMFORM:60</a></i>;<br/><b>hence </b><a NAME="E8:18_2_1"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="heyting1.html#E2:18_2_1"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="heyting1.html#E6:18_2_1"><i><font color="Green" title="E23">A13</font></i></a>, <a class="ref" href="normform.html#T4" title="NORMFORM:th.4">NORMFORM:4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:18_2"/>
<font color="Maroon" title="c4">a</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="c2">u</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="heyting1.html#E2:18_2"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="normform.html#T61" title="NORMFORM:th.61">NORMFORM:61</a></i>;<br/>


</div><b>end;</b></div>
<font color="Maroon" title="c2">u</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">v</font> = 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</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 href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
<br/>.= 
 <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="c2">u</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="normform.html#D14" title="NORMFORM:def.14">NORMFORM:def 14</a></i>
<br/>.= 
<font color="Maroon" title="c3">v</font>
<b>by </b><i><a class="txt" href="heyting1.html#E2:18"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="heyting1.html#E3:18"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>
;<br/>
<b>hence </b><a NAME="E5:18"/>
<font color="Maroon" title="c2">u</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">v</font>
 <b>by </b><i><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></i>;<br/>


</div>
