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

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K1" title="ROBBINS4:func.1">B_6</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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 <a href="robbins4.html#K2" title="ROBBINS4:func.2">Benzene</a>   st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> holds <br/>( <font color="Maroon" title="c1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">y</font> &amp; <font color="Maroon" title="c1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">y</font> )</span><br/><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 <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="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> implies ( <font color="Maroon" title="c1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">y</font> ) )</span><br/>



<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E21">X0</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">y</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c5">xy</font> = <font color="Maroon" title="c3">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K1" title="ROBBINS4:func.1">B_6</a>  <b>by </b><i><a class="ref" href="robbins4.html#T9" target="_self" title="ROBBINS4:th.9">LemBen</a>, <a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E3:28"/>
( <font color="Maroon" title="c3">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c4">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">y</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T22" title="LATTICES:th.22">LATTICES:22</a></i>;<br/>
<a NAME="E4:28"/><b>then </b><i><font color="Green" title="E22">X1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">xy</font> &amp; <font color="Maroon" title="c2">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">xy</font> )
 
<b>by </b><i><a class="txt" href="#E1:28"><i><font color="Green" title="E21">X0</font></i></a>, <a class="ref" href="robbins4.html#T13" target="_self" title="ROBBINS4:th.13">HiHi</a></i>;<br/>
<a NAME="E5:28"/><i><font color="Green" title="E23">t1</font></i>: 
for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K1" title="ROBBINS4:func.1">B_6</a>   st <font color="Olive" title="b1">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Olive" title="b1">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font> holds <br/><font color="Maroon" title="c5">xy</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K1" title="ROBBINS4:func.1">B_6</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c6">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font> implies <font color="Maroon" title="c5">xy</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">d</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c7">e</font> = <font color="Maroon" title="c6">d</font> as   <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>  <b>by </b><i><a class="ref" href="robbins4.html#T9" target="_self" title="ROBBINS4:th.9">LemBen</a>, <a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<b>assume </b><a NAME="E2:28_1"/>
( <font color="Maroon" title="c6">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c6">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">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="c5">xy</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">d</font></span><br/>

<a NAME="E3:28_1"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">e</font> &amp; <font color="Maroon" title="c4">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">e</font> )
 
<b>by </b><i><a class="txt" href="#E1:28"><i><font color="Green" title="E21">X0</font></i></a>, <a class="ref" href="robbins4.html#T13" target="_self" title="ROBBINS4:th.13">HiHi</a></i>;<br/>
<a NAME="E4:28_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">e</font>
 
<b>by </b><i><a class="ref" href="filter_0.html#T6" title="FILTER_0:th.6">FILTER_0:6</a></i>;<br/>
<b>hence </b><a NAME="E5:28_1"/>
<font color="Maroon" title="c5">xy</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="ref" href="robbins4.html#T13" target="_self" title="ROBBINS4:th.13">HiHi</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>reconsider </b><font color="Maroon" title="c6">xy</font> = <font color="Maroon" title="c3">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K1" title="ROBBINS4:func.1">B_6</a>  <b>by </b><i><a class="ref" href="robbins4.html#T9" target="_self" title="ROBBINS4:th.9">LemBen</a>, <a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E7:28"/>
( <font color="Maroon" title="c3">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c3">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">y</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T23" title="LATTICES:th.23">LATTICES:23</a></i>;<br/>
<a NAME="E8:28"/><b>then </b><i><font color="Green" title="E24">x1</font></i>: 
( <font color="Maroon" title="c6">xy</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c6">xy</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 
<b>by </b><i><a class="txt" href="#E1:28"><i><font color="Green" title="E21">X0</font></i></a>, <a class="ref" href="robbins4.html#T13" target="_self" title="ROBBINS4:th.13">HiHi</a></i>;<br/>
<a NAME="E9:28"/>
for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K1" title="ROBBINS4:func.1">B_6</a>   st <font color="Olive" title="b1">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Olive" title="b1">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">b</font> holds <br/><font color="Maroon" title="c6">xy</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins4.html#K1" title="ROBBINS4:func.1">B_6</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c7">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">b</font> implies <font color="Maroon" title="c6">xy</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">d</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c8">e</font> = <font color="Maroon" title="c7">d</font> as   <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>  <b>by </b><i><a class="ref" href="robbins4.html#T9" target="_self" title="ROBBINS4:th.9">LemBen</a>, <a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<b>assume </b><a NAME="E2:28_2"/>
( <font color="Maroon" title="c7">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c7">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">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="c6">xy</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">d</font></span><br/>

<a NAME="E3:28_2"/><b>then </b>
( <font color="Maroon" title="c8">e</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c8">e</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">y</font> )
 
<b>by </b><i><a class="txt" href="#E1:28"><i><font color="Green" title="E21">X0</font></i></a>, <a class="ref" href="robbins4.html#T13" target="_self" title="ROBBINS4:th.13">HiHi</a></i>;<br/>
<a NAME="E4:28_2"/><b>then </b>
<font color="Maroon" title="c8">e</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">y</font>
 
<b>by </b><i><a class="ref" href="filter_0.html#T7" title="FILTER_0:th.7">FILTER_0:7</a></i>;<br/>
<b>hence </b><a NAME="E5:28_2"/>
<font color="Maroon" title="c6">xy</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">d</font>
 <b>by </b><i><a class="ref" href="robbins4.html#T13" target="_self" title="ROBBINS4:th.13">HiHi</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="E10:28"/>
( <font color="Maroon" title="c1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">y</font> )
 <b>by </b><i><a class="txt" href="#E5:28"><i><font color="Green" title="E23">t1</font></i></a>, <a class="txt" href="#E4:28"><i><font color="Green" title="E22">X1</font></i></a>, <a class="txt" href="#E8:28"><i><font color="Green" title="E24">x1</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
