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<b>let </b><font color="Maroon">T</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/>

<b>assume </b><a NAME="E1:63_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">T</font> is <a href="waybel19.html#V2" target="_self">Lawson</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>, <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D4">COMPTS_1:def 4</a><br/>

<b>assume </b><a NAME="E2:63_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E4:63_1_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> holds <br/> ( not  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font> is <a href="waybel_0.html#V1">directed</a> )
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_1_1_1"/>
(  not <font color="Maroon">y'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x'</font> or  not <font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y'</font> )
 </span><b>by </b><i>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_1"><b>suppose </b></a><a NAME="E1:63_1_1_1_1"/>
not <font color="Maroon">y'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x'</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:63_1_1_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">u</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y'</font> &amp; not <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x'</font> )
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T24">WAYBEL_3:24</a></i>;<br/><b>take </b><font color="Maroon">V</font> = <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">u</font></span>)</span> <a href="subset_1.html#K3">`</a> ;<br/><b>take </b><font color="Maroon">W</font> =  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">u</font>;<br/><b>thus </b><a NAME="E4:63_1_1_1_1"/>
( <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="waybel19.html#T1" target="_self">Th1</a>, , </i>;<br/><b>thus </b><a NAME="E5:63_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="yellow_9.html#T1">YELLOW_9:1</a></i>;<br/><b>thus </b><a NAME="E6:63_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/><a NAME="E7:63_1_1_1_1"/>
( <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">u</font> &amp; <font color="Maroon">V</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">u</font> )
 <b>by </b><i><a class="ref" href="waybel_3.html#T11">WAYBEL_3:11</a></i>;<br/><b>hence </b><a NAME="E8:63_1_1_1_1"/>
<font color="Maroon">V</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_2"><b>suppose </b></a><a NAME="E1:63_1_1_1_2"/>
not <font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y'</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:63_1_1_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">u</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x'</font> &amp; not <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y'</font> )
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T24">WAYBEL_3:24</a></i>;<br/><b>take </b><font color="Maroon">W</font> =  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">u</font>;<br/><b>take </b><font color="Maroon">V</font> = <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">u</font></span>)</span> <a href="subset_1.html#K3">`</a> ;<br/><b>thus </b><a NAME="E4:63_1_1_1_2"/>
( <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="waybel19.html#T1" target="_self">Th1</a>, , <a class="txt" href="waybel19.html#E3:63_1_1_1_2"><i><font color="Green">E40</font></i></a></i>;<br/><b>thus </b><a NAME="E5:63_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:63_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="yellow_9.html#T1">YELLOW_9:1</a></i>;<br/><a NAME="E7:63_1_1_1_2"/>
( <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">u</font> &amp; <font color="Maroon">V</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">u</font> )
 <b>by </b><i><a class="ref" href="waybel_3.html#T11">WAYBEL_3:11</a></i>;<br/><b>hence </b><a NAME="E8:63_1_1_1_2"/>
<font color="Maroon">W</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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