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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#V4">antisymmetric</a> <a href="lattice3.html#V3">complete</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<a NAME="E1:29"/>
 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">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font> ) )
 
<b>by </b><i><a class="ref" href="lattice3.html#D12">LATTICE3:def 12</a></i>;<br/>
<b>hence </b><a NAME="E2:29"/>
 <a href="yellow_0.html#R1" target="_self">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/>

<b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">c</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:29"/><i><font color="Green">E24</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">c</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> </span>}</span>  <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E5:29"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">c</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> </span>}</span>  <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D12">LATTICE3:def 12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><b>thus </b><a NAME="E1:29_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="29_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="E1:29_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/>

<a NAME="E2:29_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">c</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> </span>}</span>  <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</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="E1:29_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">c</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> </span>}</span> 
 ;<br/>

<a NAME="E2:29_1_1_1"/><b>then </b>
 ex <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">d</font> &amp; <font color="Olive">d</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> )
 
;<br/>
<b>hence </b><a NAME="E3:29_1_1_1"/>
<font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:29_1_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E2:29_1"/>
<font color="Maroon">b</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E3:29_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">c</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E4:29_1"/>
<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="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:29"/>
 <a href="yellow_0.html#R2" target="_self">ex_inf_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/>


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