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<b>let </b><font color="Maroon" title="c1">FT</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font>;<br/>





<a NAME="E1:11"/><i><font color="Green" title="E6">A1</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  implies  ex <font color="Olive" title="b1">y1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> st  <a href="fintopo2.html#K1" title="FINTOPO2:func.1">P_1</a> <font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">y1</font>,<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">z</font> = <font color="Maroon" title="c2">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> ;<br/>
<a NAME="E3:11_1"/>
 <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c4">z</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="fintopo2.html#E1:11_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="fin_topo.html#T13" title="FIN_TOPO:th.13">FIN_TOPO:13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:11_1"/><i><font color="Green" title="E8">A3</font></i>: 
( <font color="Maroon" title="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c4">z</font> &amp; <font color="Maroon" title="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">w</font> = <font color="Maroon" title="c5">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> <b>by </b><i><a class="txt" href="fintopo2.html#E5:11_1"><i><font color="Green" title="E8">A3</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">w</font>
;<br/>

<b>thus </b><a NAME="E7:11_1"/>
 <a href="fintopo2.html#K1" title="FINTOPO2:func.1">P_1</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="fintopo2.html#E5:11_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="fintopo2.html#D1" target="_self" title="FINTOPO2:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:11"/>
(  ex <font color="Olive" title="b1">y1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> st  <a href="fintopo2.html#K1" title="FINTOPO2:func.1">P_1</a> <font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">y1</font>,<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c4">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font><b> such that </b><a NAME="E1:11_2"/><i><font color="Green" title="E7">A4</font></i>: 
 <a href="fintopo2.html#K1" title="FINTOPO2:func.1">P_1</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ;<br/>

<a NAME="E2:11_2"/>
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 
<b>by </b><i><a class="txt" href="fintopo2.html#E1:11_2"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="fintopo2.html#D1" target="_self" title="FINTOPO2:def.1">Def1</a></i>;<br/>
<a NAME="E3:11_2"/><b>then </b>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:11_2"/><b>then </b>
 <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c2">x</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E5:11_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> 
 <b>by </b><i><a class="ref" href="fin_topo.html#T13" title="FIN_TOPO:th.13">FIN_TOPO:13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:11"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  iff  ex <font color="Olive" title="b1">y1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> st  <a href="fintopo2.html#K1" title="FINTOPO2:func.1">P_1</a> <font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">y1</font>,<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 <b>by </b><i><a class="txt" href="fintopo2.html#E1:11"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/>


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