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<b>let </b><font color="Maroon">N</font> be  <a href="waybel_2.html#V2">meet-continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font>;<br/>



<b>assume </b><a NAME="E1:26"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">D</font> being non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font>  st  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> &amp; <font color="Olive">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">y</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D3">WAYBEL11:def 3</a><br/>

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D3">WAYBEL11:def 3</a><br/>

<b>assume </b><a NAME="E2:26"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E4:26"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">aa</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font> <b>by </b><i><a class="ref" href="waybel_0.html#T5">WAYBEL_0:5</a></i>;<br/>
<a NAME="E6:26"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="waybel_2.html#T52">WAYBEL_2:52</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E8:26"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">aa</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">D</font> &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">aa</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">D</font> &amp; <font color="Olive">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:26"/>
<font color="Maroon">aa</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">d</font></span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#T42">YELLOW_4:42</a></i>;<br/>
<b>then consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E11:26"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">d</font>
;<br/>

<b>thus </b><a NAME="E12:26"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/>

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

<b>assume </b><a NAME="E13:26"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">d</font> )
 ;<br/>

<a NAME="E14:26"/>
<font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#T24">ORDERS_2:24</a></i>;<br/>
<a NAME="E15:26"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E16:26"/>
<font color="Maroon">d</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#T23">YELLOW_0:23</a></i>;<br/>
<a NAME="E17:26"/><b>then </b>
<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="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<b>hence </b><a NAME="E18:26"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/>


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