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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#V3" title="STRUCT_0:attr.3">empty</a>  <a href="robbins1.html#L2" title="ROBBINS1:struct.2">ComplLattStr</a> ;<br/>

<b>assume </b><a NAME="E1:47"/><i><font color="Green" title="E22">A1</font></i>: 
(  <a href="robbins1.html#G2" title="ROBBINS1:aggr.2">ComplLattStr</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="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#G2" title="ROBBINS1:aggr.2">ComplLattStr</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="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="robbins3.html#V9" title="ROBBINS3:attr.9">with_Top</a> )
 ;<br/>

<a NAME="E2:47"/><i><font color="Green" title="E23">A2</font></i>: 
 <a href="lattices.html#G2" title="LATTICES:aggr.2">\/-SemiLattStr</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> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#G2" title="LATTICES:aggr.2">\/-SemiLattStr</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> #)
 
<b>by </b><i><a class="txt" href="robbins3.html#E1:47"><i><font color="Green" title="E22">A1</font></i></a></i>;<br/>
<a NAME="E3:47"/>
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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_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#E1:47"><i><font color="Green" title="E22">A1</font></i></a></i>;<br/>
<a NAME="E2:47_1"/><i><font color="Green" title="E24">A3</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#E1:47"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="robbins3.html#T18" target="_self" title="ROBBINS3:th.18">Th18</a></i>;<br/>
<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> = 
<font color="Maroon" title="c5">x'</font> <a href="robbins3.html#NK1" title="ROBBINS3:NK.1">|_|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x'</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="robbins3.html#E2:47"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="robbins3.html#E2:47_1"><i><font color="Green" title="E24">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">y'</font> <a href="robbins3.html#NK1" title="ROBBINS3:NK.1">|_|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y'</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="robbins3.html#E1:47"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="robbins3.html#D7" target="_self" title="ROBBINS3:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c4">y</font> <a href="robbins3.html#NK1" title="ROBBINS3:NK.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>
<b>by </b><i><a class="txt" href="robbins3.html#E2:47"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="robbins3.html#E2:47_1"><i><font color="Green" title="E24">A3</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E4:47_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> <font color="Maroon" title="c4">y</font> <a href="robbins3.html#NK1" title="ROBBINS3:NK.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>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:47"/>
<font color="Maroon" title="c2">L</font> is <a href="robbins3.html#V9" title="ROBBINS3:attr.9">with_Top</a>
 <b>by </b><i><a class="ref" href="robbins3.html#D7" target="_self" title="ROBBINS3:def.7">Def7</a></i>;<br/>


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