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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V3">transitive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4" target="_self">full</a>  <a href="yellow_0.html#M1" target="_self">SubRelStr</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:91"/><i><font color="Green">E24</font></i>: 
 <a href="yellow_0.html#R1" target="_self">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E2:91"/><i><font color="Green">E29</font></i>: 
 <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">a</font> =  <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="yellow_0.html#T12" target="_self">Th12</a></i>;<br/>
<b>consider </b><font color="Maroon">a'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:91"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">a'</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Olive">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a'</font> ) )
 <b>and </b><br/><a NAME="E6:91"/>
for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">c</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Olive">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">c</font> ) holds <br/><font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T11" target="_self">Th11</a>, </i>;<br/>
<a NAME="E7:91"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T11" target="_self">Th11</a>, <a class="ref" href="yellow_0.html#D4" target="_self">Def4</a>, </i>;<br/>
<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>now </b></a><div class="add"><a NAME="E1:91_1"/>
 <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T11" target="_self">Th11</a>, </i>;<br/><b>hence </b><a NAME="E2:91_1"/>
<font color="Maroon">a</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T6" target="_self">Th6</a></i>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="yellow_0.html#T4" target="_self">Th4</a></i>;<br/><b>assume </b><a NAME="E4:91_1"/>
<font color="Maroon">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E5:91_1"/><b>then </b>
<font color="Maroon">b'</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E6:91_1"/><b>then </b>
<font color="Maroon">b'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T11" target="_self">Th11</a>, </i>;<br/><b>hence </b><a NAME="E7:91_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E9:91"/>
 <a href="yellow_0.html#R1" target="_self">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">a</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D7" target="_self">YELLOW_0:def 7</a><br/>

<b>thus </b><a NAME="E1:91_2"/>
( <font color="Maroon">a</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st <font color="Olive">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font> holds <br/><font color="Olive">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font> ) )
 <b>by </b><i/>;<br/>

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

<b>reconsider </b><font color="Maroon">c'</font> = <font color="Maroon">c</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="yellow_0.html#T4" target="_self">Th4</a></i>;<br/>
<b>assume </b><a NAME="E3:91_2"/>
<font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c</font>
 ;<br/>

<a NAME="E4:91_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c'</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T7" target="_self">Th7</a></i>;<br/>
<b>assume </b><a NAME="E5:91_2"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Olive">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font>
 ;<br/>

<a NAME="E6:91_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:91_2_1"/>
<font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:91_2_1"/><b>then </b>
( <font color="Maroon">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">a'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c'</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#D4" target="_self">Def4</a>, <a class="ref" href="yellow_0.html#D5" target="_self">Def5</a>, <a class="ref" href="yellow_0.html#D8" target="_self">Def8</a>, <a class="ref" href="yellow_0.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E3:91_2_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c'</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:91_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T11" target="_self">Th11</a>, <a class="ref" href="yellow_0.html#D7" target="_self">Def7</a>, </i>;<br/>


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

<b>hence </b><a NAME="E10:91"/>
 <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/>


</div>
