<?xml version="1.0"?>
<div class="add">

<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="E1:10"/><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#K8">^i</a>  implies for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  st  <a href="fintopo2.html#K3" target="_self">P_0</a> <font color="Maroon">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  holds <br/> <a href="fintopo2.html#K1" target="_self">P_1</a> <font color="Maroon">x</font>,<font color="Olive">y</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="10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_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#K8">^i</a> 
 ;<br/>

<b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>;<br/>

<b>assume </b><a NAME="E2:10_1"/><i><font color="Green">E23</font></i>: 
 <a href="fintopo2.html#K3" target="_self">P_0</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>

<a NAME="E3:10_1"/><i><font color="Green">E33</font></i>: 
 <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="fin_topo.html#T12">FIN_TOPO:12</a></i>;<br/>
<a NAME="E4:10_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:10_1"/>
 <a href="fintopo2.html#K1" target="_self">P_1</a> <font color="Maroon">x</font>,<font color="Maroon">y</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="E2:10"/>
( ( for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  st  <a href="fintopo2.html#K3" target="_self">P_0</a> <font color="Maroon">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  holds <br/> <a href="fintopo2.html#K1" target="_self">P_1</a> <font color="Maroon">x</font>,<font color="Olive">y</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#K8">^i</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_2"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  st  <a href="fintopo2.html#K3" target="_self">P_0</a> <font color="Maroon">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  holds <br/> <a href="fintopo2.html#K1" target="_self">P_1</a> <font color="Maroon">x</font>,<font color="Olive">y</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>

<a NAME="E2:10_2"/>
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font> holds <br/>( <font color="Olive">y</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="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>;<br/>

<b>assume </b><a NAME="E1:10_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:10_2_1"/><b>then </b>
 <a href="fintopo2.html#K3" target="_self">P_0</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:10_2_1"/><b>then </b>
 <a href="fintopo2.html#K1" target="_self">P_1</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:10_2_1"/>
( <font color="Maroon">y</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">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:10_2"/><b>then </b>
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font> holds <br/><font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
;<br/>
<a NAME="E4:10_2"/><b>then </b>
 <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>
<b>hence </b><a NAME="E5:10_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="fin_topo.html#K8">^i</a> 
 <b>by </b><i><a class="ref" href="fin_topo.html#T12">FIN_TOPO:12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="fin_topo.html#K8">^i</a>  iff for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  st  <a href="fintopo2.html#K3" target="_self">P_0</a> <font color="Maroon">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  holds <br/> <a href="fintopo2.html#K1" target="_self">P_1</a> <font color="Maroon">x</font>,<font color="Olive">y</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>
