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<div class="add">

<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#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:141_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">L</font> is <a href="lattice3.html#V3">complete</a>
 ;<br/>

<b>thus </b><a NAME="E2:141_1_1"/>
<font color="Maroon">L</font> is <a href="waybel_0.html#V24" target="_self">up-complete</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_1_1"><b>proof </b></a><div class="add">

<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/>

<b>thus </b><a NAME="E1:141_1_1_1"/>
 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> ) )
 <b>by </b><i><a class="ref" href="waybel_0.html#D11" target="_self">Def11</a>, <a class="ref" href="lattice3.html#D12">LATTICE3:def 12</a></i>;<br/>


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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</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#D40" target="_self">WAYBEL_0:def 40</a><br/>

<b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> </span>}</span> ;<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="E4:141_1_1"/><i><font color="Green">E30</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y</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">x</font>
 <b>and </b><br/><a NAME="E5:141_1_1"/><i><font color="Green">E31</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">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y</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">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D11" target="_self">Def11</a>, <a class="ref" href="lattice3.html#D12">LATTICE3:def 12</a></i>;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E6:141_1_1"/>
<font color="Maroon">x</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="141_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:141_1_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:141_1_1_2"/>
<font color="Maroon">y</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:141_1_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> </span>}</span> 
 ;<br/>

<a NAME="E2:141_1_1_2_1"/><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">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> )
 
;<br/>
<b>hence </b><a NAME="E3:141_1_1_2_1"/>
<font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</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="E3:141_1_1_2"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/>


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

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

<b>assume </b><a NAME="E7:141_1_1"/>
<font color="Maroon">y</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E8:141_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E9:141_1_1"/>
<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="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>


</div>
