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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">B</font> be   <a href="lattices.html#NM8">B_Lattice</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>;<br/>





<b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:54"/>
( <font color="Maroon">X</font> <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Maroon">a</font> implies <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span>  <a href="lattice3.html#NR7" target="_self">is_great_than</a> <font color="Maroon">a</font> <a href="lattices.html#K7">`</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_1"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17" target="_self">LATTICE3:def 17</a><br/>

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</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="E2:54_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E4:54_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="lattices.html#K7">`</a>  &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>
<a NAME="E5:54_1"/>
<font color="Maroon">c</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:54_1"/>
<font color="Maroon">a</font> <a href="lattices.html#K7">`</a>  <a href="lattices.html#R1">[=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="lattices.html#T53">LATTICES:53</a></i>;<br/>


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

<b>assume </b><a NAME="E2:54"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span>  holds <br/><font color="Maroon">a</font> <a href="lattices.html#K7">`</a>  <a href="lattices.html#R3">[=</a> <font color="Olive">b</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16" target="_self">LATTICE3:def 16</a><br/>

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17" target="_self">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E3:54"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E4:54"/><b>then </b>
<font color="Maroon">b</font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 
;<br/>
<a NAME="E5:54"/><b>then </b>
( <font color="Maroon">a</font> <a href="lattices.html#K7">`</a>  <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font> <a href="lattices.html#K7">`</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="lattices.html#T49">LATTICES:49</a></i>;<br/>
<b>hence </b><a NAME="E6:54"/>
<font color="Maroon">b</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="lattices.html#T53">LATTICES:53</a></i>;<br/>


</div>
