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

<b>let </b><font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c2">L2</font> be    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E17">A1</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">L2</font> #)
 ;<br/>

<b>let </b><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E2:26"/>
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L1</font> implies  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">L2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c4">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font><b> such that </b><a NAME="E1:26_1"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c4">a</font>
 <b>and </b><br/><a NAME="E2:26_1"/><i><font color="Green" title="E19">A3</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  st <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">a</font>
 <b>and </b><br/><a NAME="E3:26_1"/><i><font color="Green" title="E20">A4</font></i>: 
for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  st <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">c</font> &amp; ( for <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  st <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b2">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">c</font> ) holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D7" target="_self" title="YELLOW_0:def.7">YELLOW_0:def 7</a><br/>

<b>reconsider </b><font color="Maroon" title="c5">a'</font> = <font color="Maroon" title="c4">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font> <b>by </b><i><a class="txt" href="yellow_0.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">a'</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D7" target="_self" title="YELLOW_0:def.7">YELLOW_0:def 7</a><br/>

<b>thus </b><a NAME="E5:26_1"/>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c5">a'</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="yellow_0.html#E1:26_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="yellow_0.html#T2" target="_self" title="YELLOW_0:th.2">Th2</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c6">b'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>;<br/><b>reconsider </b><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c6">b'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> <b>by </b><i><a class="txt" href="yellow_0.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/><b>assume </b><a NAME="E2:26_1_1"/>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c6">b'</font>
 ;<br/><a NAME="E3:26_1_1"/><b>then </b>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="yellow_0.html#T2" target="_self" title="YELLOW_0:th.2">Th2</a></i>;<br/><a NAME="E4:26_1_1"/><b>then </b>
<font color="Maroon" title="c7">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E2:26_1"><i><font color="Green" title="E19">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E5:26_1_1"/>
<font color="Maroon" title="c6">b'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">a'</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="yellow_0.html#T1" target="_self" title="YELLOW_0:th.1">Th1</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c6">c'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c7">c</font> = <font color="Maroon" title="c6">c'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> <b>by </b><i><a class="txt" href="yellow_0.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/>
<b>assume </b><a NAME="E8:26_1"/>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c6">c'</font>
 ;<br/>

<a NAME="E9:26_1"/><b>then </b><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c7">c</font>
 
<b>by </b><i><a class="txt" href="yellow_0.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="yellow_0.html#T2" target="_self" title="YELLOW_0:th.2">Th2</a></i>;<br/>
<b>assume </b><a NAME="E10:26_1"/><i><font color="Green" title="E22">A6</font></i>: 
for <font color="Olive" title="b1">b'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>  st <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b'</font> holds <br/><font color="Olive" title="b1">b'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">c'</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>;<br/><b>reconsider </b><font color="Maroon" title="c9">b'</font> = <font color="Maroon" title="c8">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font> <b>by </b><i><a class="txt" href="yellow_0.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/><b>assume </b><a NAME="E2:26_1_2"/>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c8">b</font>
 ;<br/><a NAME="E3:26_1_2"/><b>then </b>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c9">b'</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="yellow_0.html#T2" target="_self" title="YELLOW_0:th.2">Th2</a></i>;<br/><a NAME="E4:26_1_2"/><b>then </b>
<font color="Maroon" title="c9">b'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">c'</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E10:26_1"><i><font color="Green" title="E22">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E5:26_1_2"/>
<font color="Maroon" title="c8">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">c</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="yellow_0.html#T1" target="_self" title="YELLOW_0:th.1">Th1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:26_1"/>
<font color="Maroon" title="c6">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a'</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E3:26_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="yellow_0.html#E9:26_1"><i><font color="Green" title="E21">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon" title="c4">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font><b> such that </b><a NAME="E3:26"/><i><font color="Green" title="E18">A7</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c4">a</font>
 <b>and </b><br/><a NAME="E4:26"/><i><font color="Green" title="E19">A8</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  st <font color="Maroon" title="c3">X</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">a</font>
 <b>and </b><br/><a NAME="E5:26"/><i><font color="Green" title="E20">A9</font></i>: 
for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  st <font color="Maroon" title="c3">X</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">c</font> &amp; ( for <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  st <font color="Maroon" title="c3">X</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b2">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">c</font> ) holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D8" target="_self" title="YELLOW_0:def.8">YELLOW_0:def 8</a><br/>

<b>reconsider </b><font color="Maroon" title="c5">a'</font> = <font color="Maroon" title="c4">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font> <b>by </b><i><a class="txt" href="yellow_0.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">a'</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D8" target="_self" title="YELLOW_0:def.8">YELLOW_0:def 8</a><br/>

<b>thus </b><a NAME="E7:26"/>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c5">a'</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="yellow_0.html#E3:26"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="yellow_0.html#T2" target="_self" title="YELLOW_0:th.2">Th2</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c6">b'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>;<br/><b>reconsider </b><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c6">b'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> <b>by </b><i><a class="txt" href="yellow_0.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/><b>assume </b><a NAME="E2:26_2"/>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c6">b'</font>
 ;<br/><a NAME="E3:26_2"/><b>then </b>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="yellow_0.html#T2" target="_self" title="YELLOW_0:th.2">Th2</a></i>;<br/><a NAME="E4:26_2"/><b>then </b>
<font color="Maroon" title="c7">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E4:26"><i><font color="Green" title="E19">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E5:26_2"/>
<font color="Maroon" title="c6">b'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">a'</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="yellow_0.html#T1" target="_self" title="YELLOW_0:th.1">Th1</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c6">c'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c7">c</font> = <font color="Maroon" title="c6">c'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> <b>by </b><i><a class="txt" href="yellow_0.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/>
<b>assume </b><a NAME="E10:26"/>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c6">c'</font>
 ;<br/>

<a NAME="E11:26"/><b>then </b><i><font color="Green" title="E21">A10</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c7">c</font>
 
<b>by </b><i><a class="txt" href="yellow_0.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="yellow_0.html#T2" target="_self" title="YELLOW_0:th.2">Th2</a></i>;<br/>
<b>assume </b><a NAME="E12:26"/><i><font color="Green" title="E22">A11</font></i>: 
for <font color="Olive" title="b1">b'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>  st <font color="Maroon" title="c3">X</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">b'</font> holds <br/><font color="Olive" title="b1">b'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">c'</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>;<br/><b>reconsider </b><font color="Maroon" title="c9">b'</font> = <font color="Maroon" title="c8">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font> <b>by </b><i><a class="txt" href="yellow_0.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/><b>assume </b><a NAME="E2:26_3"/>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c8">b</font>
 ;<br/><a NAME="E3:26_3"/><b>then </b>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c9">b'</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="yellow_0.html#T2" target="_self" title="YELLOW_0:th.2">Th2</a></i>;<br/><a NAME="E4:26_3"/><b>then </b>
<font color="Maroon" title="c9">b'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">c'</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E12:26"><i><font color="Green" title="E22">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E5:26_3"/>
<font color="Maroon" title="c8">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">c</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="yellow_0.html#T1" target="_self" title="YELLOW_0:th.1">Th1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:26"/>
<font color="Maroon" title="c6">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a'</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E5:26"><i><font color="Green" title="E20">A9</font></i></a>, <a class="txt" href="yellow_0.html#E11:26"><i><font color="Green" title="E21">A10</font></i></a></i>;<br/>


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