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<b>let </b><font color="Maroon">L</font> be    <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">R</font> be  <a href="waybel_4.html#V1">auxiliary(i)</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">C</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:50"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>and </b><br/><a NAME="E2:50"/><i><font color="Green">E31</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">x</font>,<font color="Maroon">L</font>
 ;<br/>

<a NAME="E3:50"/><i><font color="Green">E32</font></i>: 
 <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">x</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:50"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">x</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">a</font> holds <br/><font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:50_1"/><i><font color="Green">E46</font></i>: 
 <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">x</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:50_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:50_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:50"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="waybel35.html#E2"><i><font color="Green">Lemma26</font></i></a>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>


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