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<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM8" title="LATTICES:NM.8">B_Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">X</font> <a href="boolealg.html#K1" title="BOOLEALG:func.1">\</a> <font color="Olive" title="b2">Y</font> <a href="boolealg.html#R1" title="BOOLEALG:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> iff <font color="Olive" title="b1">X</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">Y</font> )</span><br/><b>let </b><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> <a href="boolealg.html#K1" title="BOOLEALG:func.1">\</a> <font color="Maroon" title="c3">Y</font> <a href="boolealg.html#R1" title="BOOLEALG:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> iff <font color="Maroon" title="c2">X</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">Y</font> )</span><br/>





<b>thus </b><a NAME="E1:31"/>
( <font color="Maroon" title="c2">X</font> <a href="boolealg.html#K1" title="BOOLEALG:func.1">\</a> <font color="Maroon" title="c3">Y</font> <a href="boolealg.html#R1" title="BOOLEALG:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> implies <font color="Maroon" title="c2">X</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">Y</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">Y</font> implies <font color="Maroon" title="c2">X</font> <a href="boolealg.html#K1" title="BOOLEALG:func.1">\</a> <font color="Maroon" title="c3">Y</font> <a href="boolealg.html#R1" title="BOOLEALG:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="boolealg.html#K1" title="BOOLEALG:func.1">\</a> <font color="Maroon" title="c3">Y</font> <a href="boolealg.html#R1" title="BOOLEALG:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">X</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">Y</font></span><br/>

<a NAME="E2:31_1"/>
( <font color="Maroon" title="c2">X</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="boolealg.html#R1" title="BOOLEALG:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> implies <font color="Maroon" title="c2">X</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  )
 
<b>by </b><i><a class="ref" href="lattices.html#T52" title="LATTICES:th.52">LATTICES:52</a></i>;<br/>
<b>hence </b><a NAME="E3:31_1"/>
<font color="Maroon" title="c2">X</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="txt" href="boolealg.html#E1:31_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="lattices.html#T49" title="LATTICES:th.49">LATTICES:49</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E2:31"/>
<font color="Maroon" title="c2">X</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">X</font> <a href="boolealg.html#K1" title="BOOLEALG:func.1">\</a> <font color="Maroon" title="c3">Y</font> <a href="boolealg.html#R1" title="BOOLEALG:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E3:31"/><b>then </b>
<font color="Maroon" title="c2">X</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">Y</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T27" title="LATTICES:th.27">LATTICES:27</a></i>;<br/>
<a NAME="E4:31"/><b>then </b><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T47" title="LATTICES:th.47">LATTICES:47</a></i>;<br/>
<a NAME="E5:31"/>
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">X</font> <a href="boolealg.html#K1" title="BOOLEALG:func.1">\</a> <font color="Maroon" title="c3">Y</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T41" title="LATTICES:th.41">LATTICES:41</a></i>;<br/>
<b>hence </b><a NAME="E6:31"/>
<font color="Maroon" title="c2">X</font> <a href="boolealg.html#K1" title="BOOLEALG:func.1">\</a> <font color="Maroon" title="c3">Y</font> <a href="boolealg.html#R1" title="BOOLEALG:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="boolealg.html#E4:31"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="boolealg.html#D3" target="_self" title="BOOLEALG:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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