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<div class="add">

<b>let </b><font color="Maroon">N</font> be  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="orders_2.html#NM1">Poset</a>;<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>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">N</font>;<br/>





<b>assume </b><a NAME="E1:31"/><i><font color="Green">E28</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">X</font>,<font color="Maroon">N</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a><br/>

<b>thus </b><a NAME="E2:31"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>,<font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<a NAME="E3:31"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">X</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D10">YELLOW_0:def 10</a></i>;<br/>
<a NAME="E4:31"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">X</font></span>)</span> <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T15">YELLOW_2:15</a></i>;<br/>
<a NAME="E5:31"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st <font color="Olive">b</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">X</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_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">N</font>;<br/>

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">b</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 ;<br/>

<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</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">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> <b>by </b><i/>;<br/>
<a NAME="E5:31_1"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">z</font></span>)</span> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_1.html#T64">WAYBEL_1:64</a></i>;<br/>
<a NAME="E6:31_1"/><b>then </b>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:31_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<a NAME="E8:31_1"/>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<a NAME="E9:31_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<a NAME="E10:31_1"/>
<font color="Maroon">X</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_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">N</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="E1:31_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:31_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:31_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<a NAME="E4:31_1_1"/>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<b>hence </b><a NAME="E5:31_1_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:31_1"/><b>then </b>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D10">YELLOW_0:def 10</a></i>;<br/>
<a NAME="E12:31_1"/><b>then </b>
<font color="Maroon">b</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_3.html#T2">YELLOW_3:2</a></i>;<br/>
<a NAME="E13:31_1"/><b>then </b>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>;<br/>
<b>hence </b><a NAME="E14:31_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#D18">WAYBEL_1:def 18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:31"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>


</div>
