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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> ;<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/>





<a NAME="E1:18"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c2">u</font> &amp; <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="heyting1.html#K3" title="HEYTING1:func.3">@</a> <font color="Maroon" title="c3">v</font> )
 
;<br/>
<b>assume </b><a NAME="E2:18"/>
<font color="Maroon" title="c2">u</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">v</font>
 ;<br/>

<b>then </b><i><font color="Green" title="E15">A2</font></i>: <font color="Maroon" title="c3">v</font> = 
<font color="Maroon" title="c2">u</font> <a href="lattices.html#K3" title="LATTICES:func.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/>.= 
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#K2" title="BINOP_1:func.2">.</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/>
<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>

<b>assume </b><a NAME="E4:18"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">c</font> = <font color="Maroon" title="c4">x</font> as    <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="heyting1.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="setwiseo.html#T14" title="SETWISEO:th.14">SETWISEO:14</a></i>;<br/>
<a NAME="E6:18"/>
<font color="Maroon" title="c5">c</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#E4:18"><i><font color="Green" title="E16">A3</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">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="E8:18"/><i><font color="Green" title="E17">A4</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> &amp; <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="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#T65" title="NORMFORM:th.65">NORMFORM:65</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">b</font>
;<br/>

<b>thus </b><a NAME="E9:18"/>
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c6">b</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="txt" href="heyting1.html#E3:18"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="heyting1.html#E8:18"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>


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