<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D20" target="_self">LATTICE3:def 20</a><br/>



<b>set </b><font color="Maroon">B</font> =  <a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font>;<br/>
<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:59"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">x</font> <a href="lattice3.html#NR7" target="_self">is_great_than</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E2:59"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span>  st <font color="Maroon">x</font> <a href="lattice3.html#NR7" target="_self">is_great_than</a> <font color="Olive">d</font> holds <br/><font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E3:59"/><i><font color="Green">E25</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">a'</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">c</font>
 <b>and </b><br/><a NAME="E4:59"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span>  st <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">a'</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="Olive">d</font> holds <br/><font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c</font>
 ;<br/>

<b>set </b><font color="Maroon">x'</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">e</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">y</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">e</font></span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">y'</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">e</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">e</font></span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  </span>}</span> ;<br/>
<b>set </b><font color="Maroon">z</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Olive">e</font></span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">e</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  </span>}</span> ;<br/>
<a NAME="E5:59"/><i><font color="Green">E31</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Olive">e</font></span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">e</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">e</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">e</font></span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:59_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Olive">e</font></span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">e</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">e</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">e</font></span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:59_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Olive">e</font></span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">e</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">e</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E3:59_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">e</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">e</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  )
 ;<br/>
<b>consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E5:59_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="lattices.html#K7">`</a>  &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E6:59_1_1"/>
( <font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">e</font></span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">i</font></span>)</span> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> )
 
<b>by </b><i><a class="ref" href="lattice3.html#T3" target="_self">Th3</a>, <a class="ref" href="lattice3.html#T4" target="_self">Th4</a>, <a class="ref" href="lattices.html#T51">LATTICES:51</a></i>;<br/>
<b>hence </b><a NAME="E7:59_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">e</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">e</font></span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  </span>}</span> 
 ;<br/>


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

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:59_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">e</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">e</font></span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">e</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E4:59_1"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> <a href="lattices.html#K7">`</a>  &amp; <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">e</font></span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  )
 ;<br/>
<b>consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E6:59_1"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:59_1"/>
( <font color="Maroon">i</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">e</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K7">`</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="lattice3.html#T3" target="_self">Th3</a>, <a class="ref" href="lattice3.html#T4" target="_self">Th4</a>, <a class="ref" href="lattices.html#T51">LATTICES:51</a></i>;<br/>
<b>hence </b><a NAME="E8:59_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Olive">e</font></span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">e</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:59"/><i><font color="Green">E97</font></i>: 
( <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> &amp; <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T49">LATTICES:49</a></i>;<br/>
<a NAME="E7:59"/><i><font color="Green">E98</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">e</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Maroon">a</font> <a href="lattices.html#K7">`</a> 
 
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E8:59"/><i><font color="Green">E101</font></i>: 
for <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span>  st <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">e</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Olive">d</font> holds <br/><font color="Maroon">a</font> <a href="lattices.html#K7">`</a>  <a href="lattices.html#R3">[=</a> <font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span>;<br/>

<a NAME="E1:59_2"/><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T49">LATTICES:49</a></i>;<br/>
<b>assume </b><a NAME="E2:59_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">e</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Maroon">d</font>
 ;<br/>

<a NAME="E3:59_2"/><b>then </b>
<font color="Maroon">x</font> <a href="lattice3.html#NR7" target="_self">is_great_than</a> <font color="Maroon">d</font> <a href="lattices.html#K7">`</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:59_2"/><b>then </b>
<font color="Maroon">d</font> <a href="lattices.html#K7">`</a>  <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:59_2"/>
<font color="Maroon">a</font> <a href="lattices.html#K7">`</a>  <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="lattices.html#T53">LATTICES:53</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:59"/><i><font color="Green">E117</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Olive">e</font></span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">e</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  </span>}</span>  <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Maroon">c</font> <a href="lattices.html#K7">`</a> 
 
<b>by </b><i>, <a class="ref" href="lattice3.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E10:59"/><i><font color="Green">E118</font></i>: 
for <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span>  st <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Olive">e</font></span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">e</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  </span>}</span>  <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Olive">d</font> holds <br/><font color="Maroon">c</font> <a href="lattices.html#K7">`</a>  <a href="lattices.html#R3">[=</a> <font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span>;<br/>

<a NAME="E1:59_3"/><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T49">LATTICES:49</a></i>;<br/>
<b>assume </b><a NAME="E2:59_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Olive">e</font></span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">e</font> <a href="lattices.html#K7">`</a> </span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  </span>}</span>  <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Maroon">d</font>
 ;<br/>

<a NAME="E3:59_3"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">e</font></span>)</span> where <font color="Olive">e</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font></span>)</span> : <font color="Olive">e</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">d</font> <a href="lattices.html#K7">`</a> 
 
<b>by </b><i><a class="ref" href="lattice3.html#T2" target="_self">Th2</a>, , </i>;<br/>
<a NAME="E4:59_3"/><b>then </b>
<font color="Maroon">d</font> <a href="lattices.html#K7">`</a>  <a href="lattices.html#R3">[=</a> <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:59_3"/>
<font color="Maroon">c</font> <a href="lattices.html#K7">`</a>  <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="lattices.html#T53">LATTICES:53</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:59"/>
 <a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font> is <a href="lattice3.html#V5" target="_self">\/-distributive</a>
 
<b>by </b><i/>;<br/>
<a NAME="E12:59"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c</font> <a href="lattices.html#K7">`</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K7">`</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="lattice3.html#T4" target="_self">Th4</a>, , , , , <a class="ref" href="lattices.html#T50">LATTICES:50</a></i>;<br/>
<b>hence </b><a NAME="E13:59"/>
<font color="Maroon">c</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T3" target="_self">Th3</a>, <a class="ref" href="lattices.html#T53">LATTICES:53</a></i>;<br/>


</div>
