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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="robbins1.html#L3" title="ROBBINS1:struct.3">OrthoLattStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> implies ( <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V8" title="ROBBINS3:attr.8">involutive</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V9" title="ROBBINS3:attr.9">with_Top</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins1.html#V9" title="ROBBINS1:attr.9">de_Morgan</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins1.html#V7" title="ROBBINS1:attr.7">well-complemented</a> ) )</span><br/>

<b>assume </b><a NAME="E1:65_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V8" title="ROBBINS3:attr.8">involutive</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V9" title="ROBBINS3:attr.9">with_Top</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins1.html#V9" title="ROBBINS1:attr.9">de_Morgan</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins1.html#V7" title="ROBBINS1:attr.7">well-complemented</a> )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c2">L'</font> = <font color="Maroon" title="c1">L</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  <a href="robbins1.html#L3" title="ROBBINS1:struct.3">OrthoLattStr</a>  ;<br/>
<a NAME="E3:65_1_1"/><i><font color="Green" title="E27">A1</font></i>: 
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="c2">L'</font> holds  <font color="Olive" title="b1">x</font> <a href="robbins3.html#NK1" title="ROBBINS3:NK.1">|_|</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="robbins3.html#NK1" title="ROBBINS3:NK.1">|_|</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>;<br/>
<a NAME="E4:65_1_1"/><i><font color="Green" title="E28">A2</font></i>: 
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="c2">L'</font> holds  <font color="Olive" title="b1">x</font> <a href="robbins3.html#NK2" title="ROBBINS3:NK.2">|^|</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins3.html#NK1" title="ROBBINS3:NK.1">|_|</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>;<br/>
<a NAME="E5:65_1_1"/><i><font color="Green" title="E29">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L'</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>;<br/>
<a NAME="E6:65_1_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L'</font> holds  <font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">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="c3">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c3">x</font></span><br/>

<a NAME="E1:65_1_1_1"/>
( <font color="Maroon" title="c3">x</font> <a href="robbins3.html#NK1" title="ROBBINS3:NK.1">|_|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c2">L'</font> &amp; <font color="Maroon" title="c3">x</font> <a href="robbins3.html#NK2" title="ROBBINS3:NK.2">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c2">L'</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins3.html#NK1" title="ROBBINS3:NK.1">|_|</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c2">L'</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins3.html#NK2" title="ROBBINS3:NK.2">|^|</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c2">L'</font> )
 
<b>by </b><i><a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E2:65_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</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>hence </b><a NAME="E7:65_1_1"/>
( <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V8" title="ROBBINS3:attr.8">involutive</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V9" title="ROBBINS3:attr.9">with_Top</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins1.html#V9" title="ROBBINS1:attr.9">de_Morgan</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins1.html#V7" title="ROBBINS1:attr.7">well-complemented</a> )
 <b>by </b><i><a class="txt" href="#E3:65_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="#E4:65_1_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="#E5:65_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="robbins3.html#D6" target="_self" title="ROBBINS3:def.6">Def6</a>, <a class="ref" href="robbins3.html#D7" target="_self" title="ROBBINS3:def.7">Def7</a>, <a class="ref" href="robbins1.html#D10" title="ROBBINS1:def.10">ROBBINS1:def 10</a>, <a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</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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