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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="fintopo2.html#L1" target="_self">FMT_Space_Str</a> ;<br/>

<a NAME="E1:45"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="fintopo2.html#V2" target="_self">Fo_filled</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="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> <a href="fintopo2.html#K9" target="_self">^Fob</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="fintopo2.html#V2" target="_self">Fo_filled</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>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="E2:45_1"/><i><font color="Green">E20</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>
 ;<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="E4:45_1"/>
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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:45_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:45_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo2.html#E1:45_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="fintopo2.html#D8" target="_self">Def8</a></i>;<br/>
<b>hence </b><a NAME="E3:45_1_1"/>
<font color="Maroon">c<sub>5</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="fintopo2.html#E2:45_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:45_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="fintopo2.html#K9" target="_self">^Fob</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:45"/>
( ( 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="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> <a href="fintopo2.html#K9" target="_self">^Fob</a>  ) implies <font color="Maroon">c<sub>1</sub></font> is <a href="fintopo2.html#V2" target="_self">Fo_filled</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_2"/><i><font color="Green">E19</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="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> <a href="fintopo2.html#K9" target="_self">^Fob</a> 
 ;<br/>

<b>assume </b><a NAME="E2:45_2"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="fintopo2.html#V2" target="_self">Fo_filled</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:45_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E5:45_2"/><i><font color="Green">E21</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="fintopo2.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E6:45_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_2_1"/>
<font color="Maroon">c<sub>3</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>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:45_2_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <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>2</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>thus </b><a NAME="E4:45_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo2.html#E5:45_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="fintopo2.html#E3:45_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:45_2"/>
not <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="fintopo2.html#K9" target="_self">^Fob</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_2_2"/><i><font color="Green">E23</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="fintopo2.html#K9" target="_self">^Fob</a> 
 ;<br/>

<a NAME="E2:45_2_2"/>
<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>2</sub></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/>
<b>hence </b><a NAME="E3:45_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo2.html#E4:45_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="fintopo2.html#E6:45_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="fintopo2.html#E1:45_2_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="fintopo2.html#T21" target="_self">Th21</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:45_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo2.html#E1:45_2"><i><font color="Green">E19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:45"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="fintopo2.html#V2" target="_self">Fo_filled</a> iff 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="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> <a href="fintopo2.html#K9" target="_self">^Fob</a>  )
 <b>by </b><i><a class="txt" href="fintopo2.html#E1:45"><i><font color="Green">E18</font></i></a></i>;<br/>


</div>
