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

<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#V24">up-complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>



<b>thus </b><a NAME="E1:32"/>
( <font color="Maroon">X</font> is <a href="waybel_6.html#V4" target="_self">order-generating</a> implies for <font color="Olive">l1</font>, <font color="Olive">l2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st not <font color="Olive">l2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">l1</font> holds <br/> ex <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">l1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">p</font> &amp; not <font color="Olive">l2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">p</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">X</font> is <a href="waybel_6.html#V4" target="_self">order-generating</a>
 ;<br/>

<b>let </b><font color="Maroon">l1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">l2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E2:32_1"/><i><font color="Green">E47</font></i>: 
not <font color="Maroon">l2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l1</font>
 ;<br/>

<b>consider </b><font color="Maroon">P</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:32_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">l1</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">P</font>,<font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1"/>
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/><font color="Maroon">l2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">p</font>
 ;<br/><a NAME="E2:32_1_1"/><b>then </b>
<font color="Maroon">l2</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/><b>hence </b><a NAME="E3:32_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:32_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">l2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font> )
 ;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<a NAME="E8:32_1"/>
<font color="Maroon">l1</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>
<b>hence </b><a NAME="E9:32_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">l1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font> &amp; not <font color="Maroon">l2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>


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

<b>thus </b><a NAME="E2:32"/>
( ( for <font color="Olive">l1</font>, <font color="Olive">l2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st not <font color="Olive">l2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">l1</font> holds <br/> ex <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">l1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">p</font> &amp; not <font color="Olive">l2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">p</font> ) ) implies <font color="Maroon">X</font> is <a href="waybel_6.html#V4" target="_self">order-generating</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">l1</font>, <font color="Olive">l2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st not <font color="Olive">l2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">l1</font> holds <br/> ex <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">l1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">p</font> &amp; not <font color="Olive">l2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">p</font> )
 ;<br/>

<b>let </b><font color="Maroon">l</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="waybel_6.html#D5" target="_self">WAYBEL_6:def 5</a><br/>

<b>set </b><font color="Maroon">y</font> =  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font></span>)</span>;<br/>
<b>thus </b><a NAME="E2:32_2"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<a NAME="E3:32_2"/><i><font color="Green">E53</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font></span>)</span> <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_2_1"/><i><font color="Green">E54</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">l</font>
 ;<br/><a NAME="E2:32_2_1"/>
<font color="Maroon">l</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_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="E1:32_2_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:32_2_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:32_2_1_1"/>
<font color="Maroon">l</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:32_2_1"/><b>then </b>
<font color="Maroon">l</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</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/><a NAME="E4:32_2_1"/><b>then </b><i><font color="Green">E55</font></i>: 
not  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font></span>)</span> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T30">ORDERS_2:30</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_2_1_2_1"/>
(  not  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> or  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> )
 </span><b>by </b><i>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:32_2_1_2_1_1"/>
not  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:32_2_1_2_1_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">l</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font> &amp; not  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font> )
 <b>by </b><i/>;<br/><a NAME="E4:32_2_1_2_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/><a NAME="E5:32_2_1_2_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</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#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E6:32_2_1_2_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:32_2_1_2_1_2"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_2_1_2_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:32_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:32_2"/>
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font></span>)</span>
 ;<br/>


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


</div>
