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

<b>let </b><font color="Maroon">FMT</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="fintopo2.html#L1" target="_self">FMT_Space_Str</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>, <font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>;<br/>



<b>assume </b><a NAME="E1:58"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">FMT</font> is <a href="fintopo2.html#V2" target="_self">Fo_filled</a>
 ;<br/>

<b>assume </b><a NAME="E2:58"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FMT</font> holds  <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font>
 ;<br/>

<a NAME="E3:58"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font> holds  <font color="Olive">C</font> <a href="fintopo2.html#K9" target="_self">^Fob</a>  <a href="tarski.html#R1">c=</a> <font color="Olive">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>;<br/>

<a NAME="E1:58_1"/>
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FMT</font>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="fintopo2.html#K9" target="_self">^Fob</a>  holds <br/><font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_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">FMT</font>;<br/>

<a NAME="E1:58_1_1"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="fintopo2.html#T2" target="_self">Th2</a></i>;<br/>
<b>assume </b><a NAME="E2:58_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> 
 ;<br/>

<a NAME="E3:58_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="fintopo2.html#D2" target="_self">Def2</a>, </i>;<br/>
<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:58_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>thus </b><a NAME="E6:58_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="fintopo2.html#T8" target="_self">Th8</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:58_1"/>
<font color="Maroon">C</font> <a href="fintopo2.html#K9" target="_self">^Fob</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:58"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font> holds  <font color="Olive">C</font> <a href="fintopo2.html#K9" target="_self">^Fob</a>  <a href="hidden.html#R1">=</a> <font color="Olive">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>;<br/>

<a NAME="E1:58_2"/>
( <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="fintopo2.html#K9" target="_self">^Fob</a>  &amp; <font color="Maroon">C</font> <a href="fintopo2.html#K9" target="_self">^Fob</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, <a class="ref" href="fintopo2.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>hence </b><a NAME="E2:58_2"/>
<font color="Maroon">C</font> <a href="fintopo2.html#K9" target="_self">^Fob</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:58"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font> <a href="fintopo2.html#K9" target="_self">^Fob</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B</font> <a href="fintopo2.html#K9" target="_self">^Fob</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 
;<br/>
<b>hence </b><a NAME="E6:58"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span>
 ;<br/>


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