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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>set </b><font color="Maroon">L</font> =  <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>;<br/>
<a NAME="E1:64"/><i><font color="Green">E34</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="yellow_1.html#D1">YELLOW_1:def 1</a></i>;<br/>
<b>let </b><font color="Maroon">x</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> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span>, <font color="Maroon">y</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> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span>;<br/>

<b>assume </b><a NAME="E2:64"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">F</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>  st <font color="Olive">F</font> is <a href="tops_2.html#V1">open</a> &amp; <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Olive">F</font> holds <br/> ex <font color="Olive">G</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Olive">F</font> st <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Olive">G</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">I</font> be   <a href="waybel_0.html#NM3">Ideal</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span>;<br/><a NAME="E1:64_1"/>
<font color="Maroon">I</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">I</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/><b>assume </b><a NAME="E3:64_1"/>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E4:64_1"/><b>then </b>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E5:64_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T22">YELLOW_1:22</a>, <a class="ref" href="yellow_1.html#T25">YELLOW_1:25</a></i>;<br/><b>then consider </b><font color="Maroon">G</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E7:64_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#D1" target="_self">Def1</a></i>;<br/><b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>consider </b><font color="Maroon">z</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> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span><b> such that </b><br/><a NAME="E10:64_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">z</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#T1">WAYBEL_0:1</a></i>;<br/><a NAME="E11:64_1"/>
(  <a href="tarski.html#K3">union</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">G</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">G</font> )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a>, <a class="ref" href="yellow_1.html#T22">YELLOW_1:22</a></i>;<br/><a NAME="E12:64_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">G</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D19">WAYBEL_0:def 19</a>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>hence </b><a NAME="E13:64_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D19">WAYBEL_0:def 19</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:64"/>
<font color="Maroon">x</font> <a href="waybel_3.html#R1" target="_self">is_way_below</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/>


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