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<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">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:21_1"/>
<font color="Maroon">A</font> is <a href="fin_topo.html#V3">open</a>
 ;<br/><a NAME="E2:21_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</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#D16">FIN_TOPO:def 16</a></i>;<br/><a NAME="E3:21_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:21_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </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><br/><a NAME="E3:21_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp;  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:21_1_1"/>
 <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/>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E4:21_1"/>
for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  st  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> holds <br/><font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:21_1"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:21"/><i><font color="Green">E41</font></i>: 
( ( for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  st  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> holds <br/><font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> ) )
 ;<br/>

<a NAME="E3:21"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> :  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">u</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> ;<br/>
<a NAME="E3:21_2"/>
 <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:21_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> :  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:21"/>
<span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> :  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> :  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> </span>}</span> 
 ;<br/>

<b>then consider </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><br/><a NAME="E3:21_3"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp;  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 ;<br/>
<b>thus </b><a NAME="E4:21_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:21"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="fin_topo.html#K8">^i</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E6:21"/>
<font color="Maroon">A</font> is <a href="fin_topo.html#V3">open</a>
 <b>by </b><i><a class="ref" href="fin_topo.html#D16">FIN_TOPO:def 16</a></i>;<br/>


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