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<b>let </b><font color="Maroon">FT</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>;<br/>





<a NAME="E1:8"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="fin_topo.html#K5">^delta</a>  implies  ex <font color="Olive">y1</font>, <font color="Olive">y2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> st <br/>(  <a href="fintopo2.html#K1" target="_self">P_1</a> <font color="Maroon">x</font>,<font color="Olive">y1</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp;  <a href="fintopo2.html#K2" target="_self">P_2</a> <font color="Maroon">x</font>,<font color="Olive">y2</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="fin_topo.html#K5">^delta</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> ;<br/>
<a NAME="E3:8_1"/><i><font color="Green">E23</font></i>: 
(  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">z</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> &amp;  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">z</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i>, <a class="ref" href="fin_topo.html#T10">FIN_TOPO:10</a></i>;<br/>
<b>then consider </b><font color="Maroon">w1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:8_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">z</font> &amp; <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">w1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">w1</font>
;<br/>

<b>consider </b><font color="Maroon">w2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:8_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">z</font> &amp; <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>take </b>
<font color="Maroon">w2</font>
;<br/>

<b>thus </b><a NAME="E9:8_1"/>
( <font color="Maroon">w2</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> &amp;  <a href="fintopo2.html#K1" target="_self">P_1</a> <font color="Maroon">x</font>,<font color="Maroon">w1</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp;  <a href="fintopo2.html#K2" target="_self">P_2</a> <font color="Maroon">x</font>,<font color="Maroon">w2</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:8"/>
(  ex <font color="Olive">y1</font>, <font color="Olive">y2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> st <br/>(  <a href="fintopo2.html#K1" target="_self">P_1</a> <font color="Maroon">x</font>,<font color="Olive">y1</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp;  <a href="fintopo2.html#K2" target="_self">P_2</a> <font color="Maroon">x</font>,<font color="Olive">y2</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="fin_topo.html#K5">^delta</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">y1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>, <font color="Maroon">y2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font><b> such that </b><a NAME="E1:8_2"/><i><font color="Green">E36</font></i>: 
 <a href="fintopo2.html#K1" target="_self">P_1</a> <font color="Maroon">x</font>,<font color="Maroon">y1</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>and </b><br/><a NAME="E2:8_2"/><i><font color="Green">E37</font></i>: 
 <a href="fintopo2.html#K2" target="_self">P_2</a> <font color="Maroon">x</font>,<font color="Maroon">y2</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>

<a NAME="E3:8_2"/>
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:8_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:8_2"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E6:8_2"/>
( <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:8_2"/><b>then </b>
 <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E8:8_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="fin_topo.html#K5">^delta</a> 
 <b>by </b><i>, <a class="ref" href="fin_topo.html#T10">FIN_TOPO:10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:8"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="fin_topo.html#K5">^delta</a>  iff  ex <font color="Olive">y1</font>, <font color="Olive">y2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> st <br/>(  <a href="fintopo2.html#K1" target="_self">P_1</a> <font color="Maroon">x</font>,<font color="Olive">y1</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp;  <a href="fintopo2.html#K2" target="_self">P_2</a> <font color="Maroon">x</font>,<font color="Olive">y2</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) )
 <b>by </b><i/>;<br/>


</div>
