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<b>let </b><font color="Maroon" title="c1">FMT</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="fintopo2.html#L1" title="FINTOPO2:struct.1">FMT_Space_Str</a> ;<br/>

<a NAME="E1:46"/><i><font color="Green" title="E19">A1</font></i>: 
( <font color="Maroon" title="c1">FMT</font> is <a href="fintopo2.html#V2" title="FINTOPO2:attr.2">Fo_filled</a> implies for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> holds  <font color="Olive" title="b1">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c1">FMT</font> is <a href="fintopo2.html#V2" title="FINTOPO2:attr.2">Fo_filled</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font>;<br/><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>



<b>assume </b><a NAME="E2:46_1"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">y</font> = <font color="Maroon" title="c3">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FMT</font> ;<br/>
<b>consider </b><font color="Maroon" title="c5">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font><b> such that </b><br/><a NAME="E5:46_1"/><i><font color="Green" title="E22">A4</font></i>: 
( <font color="Maroon" title="c5">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c5">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="fintopo2.html#E2:46_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="fintopo2.html#T22" target="_self" title="FINTOPO2:th.22">Th22</a></i>;<br/>
<a NAME="E6:46_1"/>
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">V</font> &amp; <font color="Maroon" title="c5">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> )
 
<b>by </b><i><a class="txt" href="fintopo2.html#E1:46_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="fintopo2.html#E5:46_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="fintopo2.html#D8" target="_self" title="FINTOPO2:def.8">Def8</a></i>;<br/>
<b>hence </b><a NAME="E7:46_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:46"/>
( ( for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> holds  <font color="Olive" title="b1">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> ) implies <font color="Maroon" title="c1">FMT</font> is <a href="fintopo2.html#V2" title="FINTOPO2:attr.2">Fo_filled</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_2"/><i><font color="Green" title="E20">A5</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> holds  <font color="Olive" title="b1">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font>
 ;<br/>

<b>assume </b><a NAME="E2:46_2"/>
not <font color="Maroon" title="c1">FMT</font> is <a href="fintopo2.html#V2" title="FINTOPO2:attr.2">Fo_filled</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FMT</font>, <font color="Maroon" title="c3">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font><b> such that </b><br/><a NAME="E4:46_2"/><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c3">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c2">y</font>
 <b>and </b><br/><a NAME="E5:46_2"/><i><font color="Green" title="E22">A7</font></i>: 
not <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font>
 <b>by </b><i><a class="ref" href="fintopo2.html#D8" target="_self" title="FINTOPO2:def.8">Def8</a></i>;<br/>
<a NAME="E6:46_2"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a> 
 
<b>by </b><i><a class="txt" href="fintopo2.html#E4:46_2"><i><font color="Green" title="E21">A6</font></i></a></i>;<br/>
<a NAME="E7:46_2"/><b>then </b>
not <font color="Maroon" title="c3">V</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">V</font>
 
<b>by </b><i><a class="txt" href="fintopo2.html#E5:46_2"><i><font color="Green" title="E22">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:46_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo2.html#E1:46_2"><i><font color="Green" title="E20">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:46"/>
( <font color="Maroon" title="c1">FMT</font> is <a href="fintopo2.html#V2" title="FINTOPO2:attr.2">Fo_filled</a> iff for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> holds  <font color="Olive" title="b1">A</font> <a href="fintopo2.html#K10" title="FINTOPO2:func.10">^Foi</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> )
 <b>by </b><i><a class="txt" href="fintopo2.html#E1:46"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>


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