<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon" title="c1">B</font> =  <a href="robbins4.html#K2" title="ROBBINS4:func.2">Benzene</a> ;<br/>
<a NAME="E1:45_1_1"/>
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K2" title="ROBBINS4:func.2">Benzene</a>  holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</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">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K2" title="ROBBINS4:func.2">Benzene</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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="Maroon" title="c2">a</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_1_1_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 </span><b>by </b><i><a class="ref" href="robbins4.html#T9" target="_self" title="ROBBINS4:th.9">LemBen</a>, <a class="ref" href="enumset1.html#D4" title="ENUMSET1:def.4">ENUMSET1:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_1_1_1_1"/><i><font color="Green" title="E34">x</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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="Maroon" title="c2">a</font></span><br/><div class="add"><a NAME="E2:45_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <b>by </b><i><a class="ref" href="robbins4.html#T23" target="_self" title="ROBBINS4:th.23">twU</a></i>;<br/><b>hence </b><a NAME="E3:45_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="#E1:45_1_1_1_1_1"><i><font color="Green" title="E34">x</font></i></a>, <a class="ref" href="robbins4.html#T23" target="_self" title="ROBBINS4:th.23">twU</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_1_1_1_2"/><i><font color="Green" title="E34">x</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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="Maroon" title="c2">a</font></span><br/><div class="add"><a NAME="E2:45_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 <b>by </b><i><a class="ref" href="robbins4.html#T23" target="_self" title="ROBBINS4:th.23">twU</a></i>;<br/><b>hence </b><a NAME="E3:45_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="#E1:45_1_1_1_1_2"><i><font color="Green" title="E34">x</font></i></a>, <a class="ref" href="robbins4.html#T23" target="_self" title="ROBBINS4:th.23">twU</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:45_1_1_1_1_3"/><i><font color="Green" title="E34">x</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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="Maroon" title="c2">a</font></span><br/><div class="add"><a NAME="E2:45_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="robbins4.html#T23" target="_self" title="ROBBINS4:th.23">twU</a></i>;<br/><b>hence </b><a NAME="E3:45_1_1_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="#E1:45_1_1_1_1_3"><i><font color="Green" title="E34">x</font></i></a>, <a class="ref" href="robbins4.html#T23" target="_self" title="ROBBINS4:th.23">twU</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:45_1_1_1_1_4"/><i><font color="Green" title="E34">x</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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="Maroon" title="c2">a</font></span><br/><div class="add"><a NAME="E2:45_1_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 <b>by </b><i><a class="ref" href="robbins4.html#T23" target="_self" title="ROBBINS4:th.23">twU</a></i>;<br/><b>hence </b><a NAME="E3:45_1_1_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="#E1:45_1_1_1_1_4"><i><font color="Green" title="E34">x</font></i></a>, <a class="ref" href="robbins4.html#T23" target="_self" title="ROBBINS4:th.23">twU</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1_5"><b>suppose </b></a><a NAME="E1:45_1_1_1_1_5"/><i><font color="Green" title="E34">x</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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="Maroon" title="c2">a</font></span><br/><div class="add"><a NAME="E2:45_1_1_1_1_5"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="ref" href="robbins4.html#T23" target="_self" title="ROBBINS4:th.23">twU</a></i>;<br/><b>hence </b><a NAME="E3:45_1_1_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="#E1:45_1_1_1_1_5"><i><font color="Green" title="E34">x</font></i></a>, <a class="ref" href="robbins4.html#T23" target="_self" title="ROBBINS4:th.23">twU</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1_6"><b>suppose </b></a><a NAME="E1:45_1_1_1_1_6"/><i><font color="Green" title="E34">x</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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="Maroon" title="c2">a</font></span><br/><div class="add"><a NAME="E2:45_1_1_1_1_6"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <b>by </b><i><a class="ref" href="robbins4.html#T23" target="_self" title="ROBBINS4:th.23">twU</a></i>;<br/><b>hence </b><a NAME="E3:45_1_1_1_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="#E1:45_1_1_1_1_6"><i><font color="Green" title="E34">x</font></i></a>, <a class="ref" href="robbins4.html#T23" target="_self" title="ROBBINS4:th.23">twU</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></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:45_1_1"/><i><font color="Green" title="E34">I</font></i>: 
 <a href="robbins4.html#K2" title="ROBBINS4:func.2">Benzene</a>  is <a href="robbins3.html#V8" title="ROBBINS3:attr.8">involutive</a>
 <b>by </b><i><a class="ref" href="robbins3.html#D6" title="ROBBINS3:def.6">ROBBINS3:def 6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="robbins4.html#K2" title="ROBBINS4:func.2">Benzene</a>  is <a href="robbins3.html#V9" title="ROBBINS3:attr.9">with_Top</a> &amp;  <a href="robbins4.html#K2" title="ROBBINS4:func.2">Benzene</a>  is <a href="robbins1.html#V9" title="ROBBINS1:attr.9">de_Morgan</a> )</span><br/>

<a NAME="E3:45_1_1"/><i><font color="Green" title="E35">B</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K2" title="ROBBINS4:func.2">Benzene</a>  holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K2" title="ROBBINS4:func.2">Benzene</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_1_2_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1 or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2 or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 </span><b>by </b><i><a class="ref" href="robbins4.html#T9" target="_self" title="ROBBINS4:th.9">LemBen</a>, <a class="ref" href="enumset1.html#D4" title="ENUMSET1:def.4">ENUMSET1:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:45_1_1_2_1_1"/><i><font color="Green" title="E36">x</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3</span><br/><div class="add"><a NAME="E2:45_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <b>by </b><i><a class="ref" href="robbins4.html#T23" target="_self" title="ROBBINS4:th.23">twU</a></i>;<br/><b>hence </b><a NAME="E3:45_1_1_2_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <b>by </b><i><a class="txt" href="#E1:45_1_1_2_1_1"><i><font color="Green" title="E36">x</font></i></a>, <a class="ref" href="robbins4.html#T26" target="_self" title="ROBBINS4:th.26">Bot2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:45_1_1_2_1_2"/><i><font color="Green" title="E36">x</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3</span><br/><div class="add"><a NAME="E2:45_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 <b>by </b><i><a class="ref" href="robbins4.html#T23" target="_self" title="ROBBINS4:th.23">twU</a></i>;<br/><b>hence </b><a NAME="E3:45_1_1_2_1_2"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <b>by </b><i><a class="txt" href="#E1:45_1_1_2_1_2"><i><font color="Green" title="E36">x</font></i></a>, <a class="ref" href="robbins4.html#T21" target="_self" title="ROBBINS4:th.21">Haczyk3</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_2_1_3"><b>suppose </b></a><a NAME="E1:45_1_1_2_1_3"/><i><font color="Green" title="E36">x</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3</span><br/><div class="add"><a NAME="E2:45_1_1_2_1_3"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="robbins4.html#T23" target="_self" title="ROBBINS4:th.23">twU</a></i>;<br/><b>hence </b><a NAME="E3:45_1_1_2_1_3"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <b>by </b><i><a class="txt" href="#E1:45_1_1_2_1_3"><i><font color="Green" title="E36">x</font></i></a>, <a class="ref" href="robbins4.html#T21" target="_self" title="ROBBINS4:th.21">Haczyk3</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_2_1_4"><b>suppose </b></a><a NAME="E1:45_1_1_2_1_4"/><i><font color="Green" title="E36">x</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3</span><br/><div class="add"><a NAME="E2:45_1_1_2_1_4"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 <b>by </b><i><a class="ref" href="robbins4.html#T23" target="_self" title="ROBBINS4:th.23">twU</a></i>;<br/><b>hence </b><a NAME="E3:45_1_1_2_1_4"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <b>by </b><i><a class="txt" href="#E1:45_1_1_2_1_4"><i><font color="Green" title="E36">x</font></i></a>, <a class="ref" href="robbins4.html#T22" target="_self" title="ROBBINS4:th.22">Haczyk4</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_2_1_5"><b>suppose </b></a><a NAME="E1:45_1_1_2_1_5"/><i><font color="Green" title="E36">x</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3</span><br/><div class="add"><a NAME="E2:45_1_1_2_1_5"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="ref" href="robbins4.html#T23" target="_self" title="ROBBINS4:th.23">twU</a></i>;<br/><b>hence </b><a NAME="E3:45_1_1_2_1_5"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <b>by </b><i><a class="txt" href="#E1:45_1_1_2_1_5"><i><font color="Green" title="E36">x</font></i></a>, <a class="ref" href="robbins4.html#T22" target="_self" title="ROBBINS4:th.22">Haczyk4</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_2_1_6"><b>suppose </b></a><a NAME="E1:45_1_1_2_1_6"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3</span><br/><div class="add"><b>hence </b><a NAME="E2:45_1_1_2_1_6"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <b>by </b><i><a class="ref" href="robbins4.html#T27" target="_self" title="ROBBINS4:th.27">Bot1</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></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E4:45_1_1"/>
 <a href="robbins4.html#K2" title="ROBBINS4:func.2">Benzene</a>  is <a href="robbins3.html#V9" title="ROBBINS3:attr.9">with_Top</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="robbins4.html#K2" title="ROBBINS4:func.2">Benzene</a>  is <a href="robbins1.html#V9" title="ROBBINS1:attr.9">de_Morgan</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K2" title="ROBBINS4:func.2">Benzene</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="robbins3.html#D7" title="ROBBINS3:def.7">ROBBINS3:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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="c3">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span><br/>

<a NAME="E1:45_1_1_3"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<b>by </b><i><a class="txt" href="#E3:45_1_1"><i><font color="Green" title="E35">B</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:45_1_1_3"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</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="c3">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E3:45_1_1"><i><font color="Green" title="E35">B</font></i></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>thus </b><a NAME="E5:45_1_1"/>
 <a href="robbins4.html#K2" title="ROBBINS4:func.2">Benzene</a>  is <a href="robbins1.html#V9" title="ROBBINS1:attr.9">de_Morgan</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:45_1_1_4"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K2" title="ROBBINS4:func.2">Benzene</a>   st <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b2">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K2" title="ROBBINS4:func.2">Benzene</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">b</font> implies <font color="Maroon" title="c3">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  )</span><br/>

<b>reconsider </b><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c3">b</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <b>by </b><i><a class="ref" href="robbins4.html#T11" target="_self" title="ROBBINS4:th.11">CosikX</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c5">y</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of 3 <b>by </b><i><a class="ref" href="card_1.html#T89" title="CARD_1:th.89">CARD_1:89</a></i>;<br/>
<a NAME="E3:45_1_1_4_1"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c2">a</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">x</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="robbins4.html#D4" target="_self" title="ROBBINS4:def.4">BenDef</a></i>;<br/>
<a NAME="E4:45_1_1_4_1"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c3">b</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="c7">y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="robbins4.html#D4" target="_self" title="ROBBINS4:def.4">BenDef</a></i>;<br/>
<b>assume </b><a NAME="E5:45_1_1_4_1"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">b</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">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span><br/>

<a NAME="E6:45_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">y</font>
 
<b>by </b><i><a class="ref" href="robbins4.html#T24" target="_self" title="ROBBINS4:th.24">Inkl</a></i>;<br/>
<a NAME="E7:45_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c7">y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T31" title="SUBSET_1:th.31">SUBSET_1:31</a></i>;<br/>
<b>hence </b><a NAME="E8:45_1_1_4_1"/>
<font color="Maroon" title="c3">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="#E3:45_1_1_4_1"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="#E4:45_1_1_4_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="robbins4.html#T24" target="_self" title="ROBBINS4:th.24">Inkl</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="E2:45_1_1_4"/>
 <a href="robbins4.html#K2" title="ROBBINS4:func.2">Benzene</a>  is <a href="robbins1.html#V9" title="ROBBINS1:attr.9">de_Morgan</a>
 <b>by </b><i><a class="ref" href="robbins4.html#T4" target="_self" title="ROBBINS4:th.4">theo</a>, <a class="txt" href="#E2:45_1_1"><i><font color="Green" title="E34">I</font></i></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>


</div>
