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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:15_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:15_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:15_1"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E5:15_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D7">YELLOW_0:def 7</a></i>;<br/><b>thus </b><a NAME="E6:15_1"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K8">~</a> 
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D8">YELLOW_0:def 8</a><br/>

<b>thus </b><a NAME="E1:15_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K8">~</a> 
 <b>by </b><i><a class="txt" href="yellow_7.html#E3:15_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="yellow_7.html#T8" target="_self">Th8</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>;<br/><b>assume </b><a NAME="E1:15_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:15_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="yellow_7.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:15_1_1_1"/><b>then </b>
 <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="yellow_7.html#E4:15_1"><i><font color="Green">E11</font></i></a></i>;<br/><b>hence </b><a NAME="E4:15_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K8">~</a> 
 <b>by </b><i><a class="ref" href="yellow_7.html#T2" target="_self">Th2</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>;<br/>

<b>assume </b><a NAME="E3:15_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E4:15_1_1"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_7.html#T9" target="_self">Th9</a></i>;<br/>
<b>assume </b><a NAME="E5:15_1_1"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> st <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:15_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:15_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K8">~</a> 
 <b>by </b><i><a class="ref" href="yellow_7.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E3:15_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="yellow_7.html#E5:15_1_1"><i><font color="Green">E14</font></i></a></i>;<br/><b>hence </b><a NAME="E4:15_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="yellow_7.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:15_1_1"/><b>then </b>
(  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow_7.html#E5:15_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yellow_7.html#E4:15_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:15_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K8">~</a> 
 ;<br/>


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

<b>assume </b><a NAME="E2:15"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span><b> such that </b><br/><a NAME="E4:15"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E5:15"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> st <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E6:15"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> st <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> st <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D8">YELLOW_0:def 8</a></i>;<br/>
<b>take </b>
 <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>3</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D7">YELLOW_0:def 7</a><br/>

<b>thus </b><a NAME="E7:15"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="yellow_7.html#E4:15"><i><font color="Green">E10</font></i></a>, <a class="ref" href="yellow_7.html#T9" target="_self">Th9</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:15_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:15_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K8">~</a> 
 <b>by </b><i><a class="ref" href="yellow_7.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E3:15_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="yellow_7.html#E5:15"><i><font color="Green">E11</font></i></a></i>;<br/><b>hence </b><a NAME="E4:15_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="yellow_7.html#T2" target="_self">Th2</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E9:15"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E10:15"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K8">~</a> 
 
<b>by </b><i><a class="ref" href="yellow_7.html#T8" target="_self">Th8</a></i>;<br/>
<b>assume </b><a NAME="E11:15"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>;<br/><b>assume </b><a NAME="E1:15_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:15_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow_7.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:15_3"/><b>then </b>
 <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="yellow_7.html#E11:15"><i><font color="Green">E14</font></i></a></i>;<br/><b>hence </b><a NAME="E4:15_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K8">~</a> 
 <b>by </b><i><a class="ref" href="yellow_7.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:15"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow_7.html#E6:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yellow_7.html#E10:15"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:15"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div>
