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<b>let </b><font color="Maroon" title="c1">K</font>, <font color="Maroon" title="c2">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> ;<br/>

<b>assume </b><a NAME="E1:48"/><i><font color="Green" title="E23">A1</font></i>: 
(  <a href="robbins1.html#G3" title="ROBBINS1:aggr.3">OrthoLattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">K</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">K</font>,the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <font color="Maroon" title="c1">K</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="robbins1.html#G3" title="ROBBINS1:aggr.3">OrthoLattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c2">L</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c2">L</font>,the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <font color="Maroon" title="c2">L</font> #) &amp; <font color="Maroon" title="c1">K</font> is <a href="robbins1.html#V9" title="ROBBINS1:attr.9">de_Morgan</a> )
 ;<br/>

<a NAME="E2:48"/>
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="lattices.html#K2" title="LATTICES:func.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="lattices.html#K1" title="LATTICES:func.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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">y'</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> <b>by </b><i><a class="txt" href="robbins3.html#E1:48"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>
<a NAME="E2:48_1"/><i><font color="Green" title="E24">A2</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x'</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y'</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  )
 
<b>by </b><i><a class="txt" href="robbins3.html#E1:48"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="robbins3.html#T18" target="_self" title="ROBBINS3:th.18">Th18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">z</font> = <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="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> <b>by </b><i><a class="txt" href="robbins3.html#E1:48"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>
<font color="Maroon" title="c3">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">y</font> = 
<font color="Maroon" title="c5">x'</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">y'</font>
<b>by </b><i><a class="txt" href="robbins3.html#E1:48"><i><font color="Green" title="E23">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x'</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">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="txt" href="robbins3.html#E1:48"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">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="txt" href="robbins3.html#E2:48_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="robbins3.html#E1:48"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="robbins3.html#T18" target="_self" title="ROBBINS3:th.18">Th18</a></i>
;<br/>
<b>hence </b><a NAME="E5:48_1"/>
<font color="Maroon" title="c3">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">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="Maroon" title="c3">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">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> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:48"/>
<font color="Maroon" title="c2">L</font> is <a href="robbins1.html#V9" title="ROBBINS1:attr.9">de_Morgan</a>
 <b>by </b><i><a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></i>;<br/>


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