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









<b>assume </b><a NAME="E1:82"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:82_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font>
 ;<br/><b>thus </b><a NAME="E2:82_1"/>
<font color="Maroon">a</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1"><b>proof </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>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:82_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E3:82_1_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T8" target="_self">Th8</a>, <a class="ref" href="yellow_0.html#T9" target="_self">Th9</a>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:82_1_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T5" target="_self">Th5</a></i>;<br/>


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

<b>assume </b><a NAME="E3:82"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 ;<br/>

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E4:82"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E6:82"/>
<font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T8" target="_self">Th8</a>, <a class="ref" href="yellow_0.html#T9" target="_self">Th9</a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E7:82"/>
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T5" target="_self">Th5</a></i>;<br/>


</div>
