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<b>let </b><font color="Maroon" title="c1">FT</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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="E1:18"/><i><font color="Green" title="E8">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#K12" title="FIN_TOPO:func.12">^f</a>  implies  ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> st <br/>(  <a href="fintopo2.html#K4" title="FINTOPO2:func.4">P_A</a> <font color="Olive" title="b1">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>  &amp;  <a href="fintopo2.html#K3" title="FINTOPO2:func.3">P_0</a> <font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">x</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="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/>
<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#K12" title="FIN_TOPO:func.12">^f</a> 
 ;<br/>

<b>then consider </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><br/><a NAME="E3:18_1"/><i><font color="Green" title="E9">A2</font></i>: 
( <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> &amp; <font color="Maroon" title="c2">x</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">y</font> )
 <b>by </b><i><a class="ref" href="fin_topo.html#T16" title="FIN_TOPO:th.16">FIN_TOPO:16</a></i>;<br/>
<a NAME="E4:18_1"/><i><font color="Green" title="E10">A3</font></i>: 
 <a href="fintopo2.html#K4" title="FINTOPO2:func.4">P_A</a> <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> 
 
<b>by </b><i><a class="txt" href="fintopo2.html#E3:18_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="fintopo2.html#D4" target="_self" title="FINTOPO2:def.4">Def4</a></i>;<br/>
<a NAME="E5:18_1"/>
 <a href="fintopo2.html#K3" title="FINTOPO2:func.3">P_0</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">x</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#E3:18_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="fintopo2.html#D3" target="_self" title="FINTOPO2:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E6:18_1"/>
 ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> st <br/>(  <a href="fintopo2.html#K4" title="FINTOPO2:func.4">P_A</a> <font color="Olive" title="b1">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>  &amp;  <a href="fintopo2.html#K3" title="FINTOPO2:func.3">P_0</a> <font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">x</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#E4:18_1"><i><font color="Green" title="E10">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:18"/>
(  ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> st <br/>(  <a href="fintopo2.html#K4" title="FINTOPO2:func.4">P_A</a> <font color="Olive" title="b1">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>  &amp;  <a href="fintopo2.html#K3" title="FINTOPO2:func.3">P_0</a> <font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">x</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#K12" title="FIN_TOPO:func.12">^f</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2"/>
 ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> st <br/>(  <a href="fintopo2.html#K4" title="FINTOPO2:func.4">P_A</a> <font color="Olive" title="b1">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>  &amp;  <a href="fintopo2.html#K3" title="FINTOPO2:func.3">P_0</a> <font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 ;<br/>

<b>then consider </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><br/><a NAME="E3:18_2"/><i><font color="Green" title="E9">A4</font></i>: 
(  <a href="fintopo2.html#K4" title="FINTOPO2:func.4">P_A</a> <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>  &amp;  <a href="fintopo2.html#K3" title="FINTOPO2:func.3">P_0</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">x</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="E4:18_2"/><i><font color="Green" title="E10">A5</font></i>: 
<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#E3:18_2"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="fintopo2.html#D4" target="_self" title="FINTOPO2:def.4">Def4</a></i>;<br/>
<a NAME="E5:18_2"/>
<font color="Maroon" title="c2">x</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">y</font>
 
<b>by </b><i><a class="txt" href="fintopo2.html#E3:18_2"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="fintopo2.html#D3" target="_self" title="FINTOPO2:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E6:18_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#K12" title="FIN_TOPO:func.12">^f</a> 
 <b>by </b><i><a class="txt" href="fintopo2.html#E4:18_2"><i><font color="Green" title="E10">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:18"/>
( <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#K12" title="FIN_TOPO:func.12">^f</a>  iff  ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> st <br/>(  <a href="fintopo2.html#K4" title="FINTOPO2:func.4">P_A</a> <font color="Olive" title="b1">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>  &amp;  <a href="fintopo2.html#K3" title="FINTOPO2:func.3">P_0</a> <font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">x</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#E1:18"><i><font color="Green" title="E8">A1</font></i></a></i>;<br/>


</div>
