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<b>let </b><font color="Maroon">L</font> be  <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:53"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> or  not  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> is <a href="waybel_3.html#V1">compact</a> )
 <b>and </b><br/><a NAME="E2:53"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">A</font> being non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  st  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">A</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">p</font> holds <br/> 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="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:53_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12">fininfs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:53_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K12">fininfs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="waybel_7.html#T8" target="_self">Th8</a>, <a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/><b>consider </b><font color="Maroon">Y</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span><b> such that </b><br/><a NAME="E6:53_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>assume </b><a NAME="E8:53_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E9:53_1"/><b>then </b>
<font color="Maroon">a</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T7">WAYBEL_3:7</a></i>;<br/><a NAME="E10:53_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">b</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T2">WAYBEL_3:2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1"/>
( <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">Y</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="53_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:53_1_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E4:53_1_1_1"/><i><font color="Green">E59</font></i>: 
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">p</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><a NAME="E5:53_1_1_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="waybel_7.html#T9" target="_self">Th9</a>, <a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/><a NAME="E6:53_1_1_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E7:53_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:53_1_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> )
 <b>by </b><i>, , <a class="ref" href="waybel_3.html#T1">WAYBEL_3:1</a>, <a class="ref" href="yellow_0.html#T45">YELLOW_0:45</a></i>;<br/><a NAME="E3:53_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/><b>hence </b><a NAME="E4:53_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="waybel_7.html#T9" target="_self">Th9</a>, <a class="ref" href="waybel_3.html#D2">WAYBEL_3:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:53"/>
 <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12">fininfs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#R1">misses</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


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