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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="fin_topo.html#L1" target="_self">FT_Space_Str</a> ;<br/>

<b>thus </b><a NAME="E1:28"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="fin_topo.html#V3" target="_self">symmetric</a> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="fin_topo.html#K8" target="_self">^b</a>  <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="fin_topo.html#K11" target="_self">^f</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="fin_topo.html#V3" target="_self">symmetric</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>thus </b><a NAME="E2:28_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="fin_topo.html#K8" target="_self">^b</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="fin_topo.html#K11" target="_self">^f</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></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="E1:28_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="fin_topo.html#K8" target="_self">^b</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:28_1_1"/>
 <a href="fin_topo.html#K1" target="_self">U_FT</a> <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="fin_topo.html#E1:28_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="fin_topo.html#T13" target="_self">Th13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:28_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" target="_self">U_FT</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="fin_topo.html#E5:28_1_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E7:28_1_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1" target="_self">U_FT</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="fin_topo.html#E5:28_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:28_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1" target="_self">U_FT</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="fin_topo.html#E1:28_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="fin_topo.html#D14" target="_self">Def14</a></i>;<br/>
<b>hence </b><a NAME="E9:28_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="fin_topo.html#K11" target="_self">^f</a> 
 <b>by </b><i><a class="txt" href="fin_topo.html#E7:28_1_1"><i><font color="Green">E16</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>3</sub></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="E3:28_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="fin_topo.html#K11" target="_self">^f</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:28_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1" target="_self">U_FT</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="fin_topo.html#E3:28_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="fin_topo.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E7:28_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1" target="_self">U_FT</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="fin_topo.html#E1:28_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fin_topo.html#E6:28_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="fin_topo.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E8:28_1"/><b>then </b>
 <a href="fin_topo.html#K1" target="_self">U_FT</a> <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="fin_topo.html#E6:28_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E9:28_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="fin_topo.html#K8" target="_self">^b</a> 
 ;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:28"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="fin_topo.html#K8" target="_self">^b</a>  <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="fin_topo.html#K11" target="_self">^f</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="fin_topo.html#D14" target="_self">FIN_TOPO:def 14</a><br/>

<b>assume </b><a NAME="E3:28"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1" target="_self">U_FT</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E4:28"/><b>then </b>
 <a href="fin_topo.html#K1" target="_self">U_FT</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T54">ZFMISC_1:54</a></i>;<br/>
<a NAME="E5:28"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="fin_topo.html#K8" target="_self">^b</a> 
 
;<br/>
<a NAME="E6:28"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="fin_topo.html#K11" target="_self">^f</a> 
 
<b>by </b><i><a class="txt" href="fin_topo.html#E2:28"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:28"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1" target="_self">U_FT</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="fin_topo.html#T16" target="_self">Th16</a></i>;<br/>
<b>thus </b><a NAME="E9:28"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1" target="_self">U_FT</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="fin_topo.html#E8:28"><i><font color="Green">E14</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div>
