<?xml version="1.0"?>
<div class="add">

<a NAME="E1:33_1_1"/>
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span></span>)</span> holds   <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Olive">X</font>, <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span></span>)</span>;<br/>

<b>set </b><font color="Maroon">N</font> =  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">X</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:33_1_1_1_1_1"/><b>then </b>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">X</font> is  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font>
 <b>by </b><i/>;<br/><a NAME="E3:33_1_1_1_1_1"/><b>then </b>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">N</font> =  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">X</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><a NAME="E5:33_1_1_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">X</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_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 <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span></span>)</span>; <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="E1:33_1_1_1_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:33_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>
<b>hence </b><a NAME="E3:33_1_1_1_1_1_1"/>
<font color="Maroon">N</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:33_1_1_1_1_1"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span></span>)</span>  st <font color="Maroon">X</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Maroon">N</font> <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="33_1_1_1_1_1_2"><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 <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:33_1_1_1_1_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">X</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:33_1_1_1_1_1_2"/>
for <font color="Olive">Z1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Z1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:33_1_1_1_1_1_2_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">Z1'</font> = <font color="Maroon">Z1</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span></span>)</span> ;<br/>
<a NAME="E3:33_1_1_1_1_1_2_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">Z1'</font>
 
<b>by </b><i>, <a class="ref" href="waybel_4.html#D2" target="_self">Def2</a>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:33_1_1_1_1_1_2_1"/>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:33_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>
<b>hence </b><a NAME="E4:33_1_1_1_1_1_2"/>
<font color="Maroon">N</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:33_1_1_1_1_1"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">X</font>, <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T16">YELLOW_0:16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_2"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">X</font>, <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T43">YELLOW_0:43</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:33_1_1"/>
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span> is <a href="lattice3.html#V3">complete</a>
 <b>by </b><i><a class="ref" href="yellow_2.html#T30">YELLOW_2:30</a></i>;<br/>


</div>
