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





<b>assume </b><b>that </b><br/><a NAME="E1:59"/><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="boolealg.html#R1" target="_self">=</a> <font color="Maroon">X</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E2:59"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">X</font> <a href="boolealg.html#R4" target="_self">misses</a> <font color="Maroon">X</font> <a href="lattices.html#K7">`</a> 
 <b>and </b><br/><a NAME="E3:59"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">Y</font> <a href="boolealg.html#R4" target="_self">misses</a> <font color="Maroon">Y</font> <a href="lattices.html#K7">`</a> 
 ;<br/>

<a NAME="E4:59"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">X</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="boolealg.html#R1" target="_self">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="boolealg.html#T46" target="_self">Th46</a>, </i>;<br/>
<a NAME="E5:59"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">Y</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="boolealg.html#R1" target="_self">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="boolealg.html#T50" target="_self">Th50</a>, </i>;<br/>
<a NAME="E6:59"/>
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="boolealg.html#R1" target="_self">=</a> <font color="Maroon">X</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="boolealg.html#T39" target="_self">Th39</a>, <a class="ref" href="lattices.html#D11">LATTICES:def 11</a></i>;<br/>
<b>then </b><i><font color="Green">E35</font></i>: <font color="Maroon">X</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="lattices.html#K7">`</a> </span>)</span> = 
<font color="Maroon">X</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="boolealg.html#T52" target="_self">Th52</a>, <a class="ref" href="lattices.html#T39">LATTICES:39</a></i>
<br/>.= 
<font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="lattices.html#D9">LATTICES:def 9</a></i>
;<br/>
<a NAME="E8:59"/>
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="boolealg.html#R1" target="_self">=</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="boolealg.html#T39" target="_self">Th39</a>, <a class="ref" href="boolealg.html#T52" target="_self">Th52</a>, <a class="ref" href="lattices.html#D11">LATTICES:def 11</a></i>;<br/>
<a NAME="E9:59"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="boolealg.html#R1" target="_self">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T39">LATTICES:39</a></i>;<br/>
<a NAME="E10:59"/>
<span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="boolealg.html#R1" target="_self">=</a> <font color="Maroon">Y</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="boolealg.html#T39" target="_self">Th39</a>, <a class="ref" href="lattices.html#D11">LATTICES:def 11</a></i>;<br/>
<a NAME="E11:59"/><b>then </b>
<font color="Maroon">Y</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="boolealg.html#R1" target="_self">=</a> <font color="Maroon">Y</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="boolealg.html#T53" target="_self">Th53</a>, <a class="ref" href="lattices.html#T39">LATTICES:39</a></i>;<br/>
<a NAME="E12:59"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">Y</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="boolealg.html#R1" target="_self">=</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="lattices.html#D9">LATTICES:def 9</a></i>;<br/>
<a NAME="E13:59"/>
<span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="boolealg.html#R1" target="_self">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">X</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="boolealg.html#T39" target="_self">Th39</a>, <a class="ref" href="boolealg.html#T53" target="_self">Th53</a>, <a class="ref" href="lattices.html#D11">LATTICES:def 11</a></i>;<br/>
<a NAME="E14:59"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="boolealg.html#R1" target="_self">=</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T39">LATTICES:39</a></i>;<br/>
<b>hence </b><a NAME="E15:59"/>
<font color="Maroon">X</font> <a href="boolealg.html#R1" target="_self">=</a> <font color="Maroon">Y</font> <a href="lattices.html#K7">`</a> 
 <b>by </b><i><a class="ref" href="boolealg.html#T54" target="_self">Th54</a>, <a class="ref" href="lattices.html#D9">LATTICES:def 9</a></i>;<br/>

<b>thus </b><a NAME="E16:59"/>
<font color="Maroon">Y</font> <a href="boolealg.html#R1" target="_self">=</a> <font color="Maroon">X</font> <a href="lattices.html#K7">`</a> 
 <b>by </b><i><a class="ref" href="boolealg.html#T59" target="_self">Th59</a>, <a class="ref" href="boolealg.html#T61" target="_self">Th61</a>, <a class="ref" href="lattices.html#D9">LATTICES:def 9</a></i>;<br/>


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