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<div class="add">

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="robbins1.html#L3">OrthoLattStr</a> ;<br/>

<b>assume </b><a NAME="E1:65_1_1"/>
<font color="Maroon">L</font> is <a href="realset2.html#V3">trivial</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">L'</font> = <font color="Maroon">L</font> as  non <a href="struct_0.html#V3">empty</a> <a href="realset2.html#V3">trivial</a>  <a href="robbins1.html#L3">OrthoLattStr</a>  ;<br/>
<a NAME="E3:65_1_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> holds  <font color="Olive">x</font> <a href="robbins3.html#NK1" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="robbins3.html#NK1" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="robbins1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<a NAME="E4:65_1_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> holds  <font color="Olive">x</font> <a href="robbins3.html#NK2" target="_self">|^|</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#NK1" target="_self">|_|</a> <span class="p2">(<span class="default"><font color="Olive">y</font> <a href="robbins1.html#K3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<a NAME="E5:65_1_1"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> holds  <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 
<b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<a NAME="E6:65_1_1"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> holds  <font color="Olive">x</font> <a href="robbins1.html#K3">`</a>  <a href="lattices.html#R2">is_a_complement_of</a> <font color="Olive">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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>;<br/>

<a NAME="E1:65_1_1_1"/>
( <font color="Maroon">x</font> <a href="robbins3.html#NK1" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L'</font> &amp; <font color="Maroon">x</font> <a href="robbins3.html#NK2" target="_self">|^|</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L'</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#NK1" target="_self">|_|</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L'</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#NK2" target="_self">|^|</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L'</font> )
 
<b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E2:65_1_1_1"/>
<font color="Maroon">x</font> <a href="robbins1.html#K3">`</a>  <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="lattices.html#D18">LATTICES:def 18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:65_1_1"/>
( <font color="Maroon">L</font> is <a href="robbins3.html#V8" target="_self">involutive</a> &amp; <font color="Maroon">L</font> is <a href="robbins3.html#V9" target="_self">with_Top</a> &amp; <font color="Maroon">L</font> is <a href="robbins1.html#V9">de_Morgan</a> &amp; <font color="Maroon">L</font> is <a href="robbins1.html#V7">well-complemented</a> )
 <b>by </b><i>, , <a class="ref" href="robbins3.html#T3" target="_self">Th3</a>, <a class="ref" href="robbins3.html#T4" target="_self">Th4</a>, , <a class="ref" href="robbins1.html#D10">ROBBINS1:def 10</a>, <a class="ref" href="robbins1.html#D23">ROBBINS1:def 23</a></i>;<br/>


</div>
