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<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_1.html#V11" title="WAYBEL_1:attr.11">Boolean</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>;<br/>





<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel15.html#K1" title="WAYBEL15:func.1">ATOM</a> <font color="Maroon" title="c1">L</font>
 ;<br/>

<b>thus </b><a NAME="E2:33"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> implies ( <font color="Maroon" title="c2">x</font> is <a href="waybel15.html#V1" title="WAYBEL15:attr.1">atom</a> &amp; <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">A</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ;<br/>

<b>hence </b><a NAME="E2:33_1"/>
<font color="Maroon" title="c2">x</font> is <a href="waybel15.html#V1" title="WAYBEL15:attr.1">atom</a>
 <b>by </b><i><a class="txt" href="waybel15.html#E1:33"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="waybel15.html#D2" target="_self" title="WAYBEL15:def.2">Def2</a></i>;<br/>

<a NAME="E3:33_1"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">A</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32" title="YELLOW_0:th.32">YELLOW_0:32</a></i>;<br/>
<b>hence </b><a NAME="E4:33_1"/>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="waybel15.html#E1:33_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:33"/>
( <font color="Maroon" title="c2">x</font> is <a href="waybel15.html#V1" title="WAYBEL15:attr.1">atom</a> &amp; <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">A</font> implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:33_2"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c2">x</font> is <a href="waybel15.html#V1" title="WAYBEL15:attr.1">atom</a>
 <b>and </b><br/><a NAME="E2:33_2"/><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">A</font>
 <b>and </b><br/><a NAME="E3:33_2"/><i><font color="Green" title="E29">A5</font></i>: 
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>assume </b><a NAME="E1:33_2_1"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b1">y</font></span>)</span> where <font color="Olive" title="b1">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c5">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:33_2_1"/><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y</font>
 <b>and </b><br/><a NAME="E4:33_2_1"/><i><font color="Green" title="E31">A7</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ;<br/><a NAME="E5:33_2_1"/>
<font color="Maroon" title="c5">y</font> is <a href="waybel15.html#V1" title="WAYBEL15:attr.1">atom</a>
 <b>by </b><i><a class="txt" href="waybel15.html#E1:33"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="waybel15.html#E4:33_2_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="waybel15.html#D2" target="_self" title="WAYBEL15:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E6:33_2_1"/>
<font color="Maroon" title="c4">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel15.html#E1:33_2"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="waybel15.html#E3:33_2"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="waybel15.html#E3:33_2_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="waybel15.html#E4:33_2_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="waybel15.html#T27" target="_self" title="WAYBEL15:th.27">Th27</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:33_2"/><b>then </b><i><font color="Green" title="E30">A8</font></i>: 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b1">y</font></span>)</span> where <font color="Olive" title="b1">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>
<font color="Maroon" title="c2">x</font> = 
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">A</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel15.html#E2:33_2"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="yellow_0.html#T25" title="YELLOW_0:th.25">YELLOW_0:25</a></i>
<br/>.= 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b1">y</font></span>)</span> where <font color="Olive" title="b1">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> </span>}</span> ,<font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="ref" href="waybel15.html#T26" target="_self" title="WAYBEL15:th.26">Th26</a></i>
;<br/>
<a NAME="E7:33_2"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel15.html#E5:33_2"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="yellow_0.html#T32" title="YELLOW_0:th.32">YELLOW_0:32</a></i>;<br/>
<a NAME="E8:33_2"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_5.html#T22" title="YELLOW_5:th.22">YELLOW_5:22</a></i>;<br/>
<b>hence </b><a NAME="E9:33_2"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel15.html#E1:33_2"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="waybel15.html#T23" target="_self" title="WAYBEL15:th.23">Th23</a></i>;<br/>


</div><b>end;</b></div>


</div>
