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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#V2">reflexive</a> <a href="orders_2.html#V4">antisymmetric</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:138_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">L</font> is <a href="waybel_0.html#V24" target="_self">up-complete</a>
 ;<br/><b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1" target="_self">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:138_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E4:138_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">y</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D6" target="_self">Def6</a></i>;<br/><b>thus </b><a NAME="E5:138_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="138_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">x</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="E1:138_1_1"/>
( <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_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">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">x</font> ) )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D11" target="_self">Def11</a></i>;<br/>

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

<b>assume </b><a NAME="E2:138_1_1"/>
( <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">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="Maroon">c</font> ) )
 ;<br/>

<a NAME="E3:138_1_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">c</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D11" target="_self">Def11</a></i>;<br/>
<b>hence </b><a NAME="E4:138_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


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

<b>assume </b><a NAME="E2:138"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1" target="_self">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> holds   <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">X</font>,<font color="Maroon">L</font>
 ;<br/>

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1" target="_self">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D39" target="_self">WAYBEL_0:def 39</a><br/>

<a NAME="E3:138"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:138"/><b>then </b>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">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="Olive">a</font> ) &amp; ( 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="Olive">a</font> ) )
 
<b>by </b><i><a class="ref" href="yellow_0.html#D7">YELLOW_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E5:138"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">x</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">y</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font> holds <br/><font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> ) )
 ;<br/>


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