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<div class="add">

<b>let </b><font color="Maroon">C</font> be  <a href="lattice3.html#V4" target="_self">complete</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font> : <font color="Olive">a</font> <a href="lattice3.html#NR8" target="_self">is_great_than</a> <font color="Maroon">X</font> </span>}</span> ;<br/>
<a NAME="E1:74"/><i><font color="Green">E18</font></i>: 
 <a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">C</font> <a href="lattice3.html#R3" target="_self">is_less_than</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font> : <font color="Olive">a</font> <a href="lattice3.html#NR8" target="_self">is_great_than</a> <font color="Maroon">X</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16" target="_self">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E1:74_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font> : <font color="Olive">a</font> <a href="lattice3.html#NR8" target="_self">is_great_than</a> <font color="Maroon">X</font> </span>}</span> 
 ;<br/>

<a NAME="E2:74_1"/><b>then </b>
 ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="lattice3.html#NR8" target="_self">is_great_than</a> <font color="Maroon">X</font> )
 
;<br/>
<b>hence </b><a NAME="E3:74_1"/>
 <a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">C</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:74"/>
<font color="Maroon">X</font> <a href="lattice3.html#R4" target="_self">is_less_than</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:74"/><b>then </b>
 <a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font> : <font color="Olive">a</font> <a href="lattice3.html#NR8" target="_self">is_great_than</a> <font color="Maroon">X</font> </span>}</span> 
 
;<br/>
<a NAME="E4:74"/><b>then </b>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font>  st <font color="Olive">b</font> <a href="lattice3.html#R3" target="_self">is_less_than</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font> : <font color="Olive">a</font> <a href="lattice3.html#NR8" target="_self">is_great_than</a> <font color="Maroon">X</font> </span>}</span>  holds <br/><font color="Olive">b</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:74"/>
 <a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16" target="_self">"/\"</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font> : <font color="Olive">a</font> <a href="lattice3.html#NR8" target="_self">is_great_than</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#T4" target="_self">Th4</a></i>;<br/>


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