<?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="E1:16"/><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#K10">^s</a>  implies (  <a href="fintopo2.html#K4" target="_self">P_A</a> <font color="Maroon">x</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; ( for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  holds <br/>(  not  <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>  or  not  <a href="fintopo2.html#K5" target="_self">P_e</a> <font color="Maroon">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="fin_topo.html#K10">^s</a> 
 ;<br/>

<a NAME="E2:16_1"/><b>then </b><i><font color="Green">E21</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <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#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="fin_topo.html#T14">FIN_TOPO:14</a></i>;<br/>
<a NAME="E3:16_1"/><b>then </b><i><font color="Green">E23</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E4:16_1"/>
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  holds <br/>(  not  <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>  or  not  <a href="fintopo2.html#K5" target="_self">P_e</a> <font color="Maroon">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font><b> such that </b><a NAME="E1:16_1_1"/><i><font color="Green">E33</font></i>: 
 <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>and </b><br/><a NAME="E2:16_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="fintopo2.html#K5" target="_self">P_e</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/>

<a NAME="E3:16_1_1"/><i><font color="Green">E36</font></i>: 
( <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>, </i>;<br/>
<a NAME="E4:16_1_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:16_1_1"/><b>then </b>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:16_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <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#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:16_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:16_1"/>
(  <a href="fintopo2.html#K4" target="_self">P_A</a> <font color="Maroon">x</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; ( for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  holds <br/>(  not  <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>  or  not  <a href="fintopo2.html#K5" target="_self">P_e</a> <font color="Maroon">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) ) )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:16"/>
(  <a href="fintopo2.html#K4" target="_self">P_A</a> <font color="Maroon">x</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; ( for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  holds <br/>(  not  <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>  or  not  <a href="fintopo2.html#K5" target="_self">P_e</a> <font color="Maroon">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) ) implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="fin_topo.html#K10">^s</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_2"/><i><font color="Green">E37</font></i>: 
(  <a href="fintopo2.html#K4" target="_self">P_A</a> <font color="Maroon">x</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; ( for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  holds <br/>(  not  <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>  or  not  <a href="fintopo2.html#K5" target="_self">P_e</a> <font color="Maroon">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) ) )
 ;<br/>

<a NAME="E2:16_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:16_2"/>
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  holds not <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_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/>

<a NAME="E1:16_2_1"/>
(  not  <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>  or  not  <a href="fintopo2.html#K5" target="_self">P_e</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E2:16_2_1"/><b>then </b>
(  not <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> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:16_2_1"/><b>then </b>
(  not <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> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:16_2_1"/><b>then </b>
(  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <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#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:16_2_1"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:16_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E5:16_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#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</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/>
<b>hence </b><a NAME="E6:16_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="fin_topo.html#K10">^s</a> 
 <b>by </b><i>, <a class="ref" href="fin_topo.html#T14">FIN_TOPO:14</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:16"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="fin_topo.html#K10">^s</a>  iff (  <a href="fintopo2.html#K4" target="_self">P_A</a> <font color="Maroon">x</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; ( for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  holds <br/>(  not  <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>  or  not  <a href="fintopo2.html#K5" target="_self">P_e</a> <font color="Maroon">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) ) ) )
 <b>by </b><i/>;<br/>


</div>
