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<b>let </b><font color="Maroon">N</font> be  <a href="orders_2.html#V4">antisymmetric</a> <a href="lattice3.html#V2">with_infima</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font>, <font color="Maroon">E</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font>;<br/><b>let </b><font color="Maroon">X</font> be  <a href="waybel_0.html#V13">upper</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font>;<br/>





<b>assume </b><a NAME="E1:24"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">D</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<b>assume </b><a NAME="E2:24"/>
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">E</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:24"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">E</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> <b>by </b><i/>;<br/>
<a NAME="E6:24"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">D</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">d</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">e</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">e</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> &amp; <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#D4">YELLOW_4:def 4</a></i>;<br/>
<a NAME="E7:24"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">E</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</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>, <font color="Maroon">e</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E9:24"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">e</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> )
 <b>by </b><i/>;<br/>
<a NAME="E10:24"/>
<font color="Maroon">d</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">e</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<a NAME="E11:24"/><b>then </b>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/>
<b>hence </b><a NAME="E12:24"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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