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

<span class="kw">let </span><font color="Maroon" title="c1">ET</font> be   <a href="fintopo7.html#NM3" title="FINTOPO7:NM.3">FMT_TopSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ET</font><br/>  for <font color="Olive" title="b2">V</font> being    <a href="fintopo7.html#M1" title="FINTOPO7:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">a</font>  ex <font color="Olive" title="b3">O</font> being   <a href="fintopo7.html#V1" title="FINTOPO7:attr.1">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">ET</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">O</font> &amp; <font color="Olive" title="b3">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ET</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being    <a href="fintopo7.html#M1" title="FINTOPO7:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">a</font>  ex <font color="Olive" title="b2">O</font> being   <a href="fintopo7.html#V1" title="FINTOPO7:attr.1">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">ET</font> st <br/>( <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">O</font> &amp; <font color="Olive" title="b2">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">V</font> be    <a href="fintopo7.html#M1" title="FINTOPO7:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">a</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">O</font> being   <a href="fintopo7.html#V1" title="FINTOPO7:attr.1">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">ET</font> st <br/>( <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">O</font> &amp; <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">V</font> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">O</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ET</font> : <font color="Maroon" title="c3">V</font> is    <a href="fintopo7.html#M1" title="FINTOPO7:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">x</font> </span> } </span> ;<br/>
<a NAME="E1:29"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ET</font> : <font color="Maroon" title="c3">V</font> is    <a href="fintopo7.html#M1" title="FINTOPO7:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">x</font> </span> } </span>  is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">ET</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:29_1"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ET</font> : <font color="Maroon" title="c3">V</font> is    <a href="fintopo7.html#M1" title="FINTOPO7:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">x</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ET</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ET</font> : <font color="Maroon" title="c3">V</font> is    <a href="fintopo7.html#M1" title="FINTOPO7:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">x</font> </span> } </span>  or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ET</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ET</font> : <font color="Maroon" title="c3">V</font> is    <a href="fintopo7.html#M1" title="FINTOPO7:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">x</font> </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ET</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">t</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ET</font><span class="kw"> such that </span><br/><a NAME="E3:29_1_1"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t</font>
 <span class="kw">and </span><br/><a NAME="E4:29_1_1"/>
<font color="Maroon" title="c3">V</font> is    <a href="fintopo7.html#M1" title="FINTOPO7:mode.1">a_neighborhood</a> of <font color="Maroon" title="c6">t</font>
 ;<br/>
<span class="kw">thus </span><a NAME="E5:29_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ET</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:29_1"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ET</font> : <font color="Maroon" title="c3">V</font> is    <a href="fintopo7.html#M1" title="FINTOPO7:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">x</font> </span> } </span>  is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">ET</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">O</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ET</font> : <font color="Maroon" title="c3">V</font> is    <a href="fintopo7.html#M1" title="FINTOPO7:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">x</font> </span> } </span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">ET</font> ;<br/>
<a NAME="E3:29"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon" title="c5">O</font> is   <a href="fintopo7.html#V1" title="FINTOPO7:attr.1">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">ET</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:29_2"/>
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ET</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">O</font> holds <br/><font color="Maroon" title="c5">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ET</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">O</font> implies <font color="Maroon" title="c5">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c6">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">O</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c6">x</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">t</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ET</font><span class="kw"> such that </span><br/><a NAME="E3:29_2_1"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font>
 <span class="kw">and </span><br/><a NAME="E4:29_2_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<font color="Maroon" title="c3">V</font> is    <a href="fintopo7.html#M1" title="FINTOPO7:mode.1">a_neighborhood</a> of <font color="Maroon" title="c7">t</font>
 ;<br/>
<a NAME="E5:29_2_1"/>
( <font color="Maroon" title="c6">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ET</font> &amp; <font color="Maroon" title="c3">V</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c6">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_2_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E4:29_2_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="fintopo7.html#D5" target="_self" onmouseover="rs('fintopo7/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">W</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c6">x</font><span class="kw"> such that </span><br/><a NAME="E7:29_2_1"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
 for <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ET</font>  st <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">W</font> holds <br/><font color="Maroon" title="c3">V</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo7.html#D4" target="_self" onmouseover="rs('fintopo7/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E8:29_2_1"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
<font color="Maroon" title="c8">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">O</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">W</font> or <font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">O</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_2_1_1"/><span class="lab"><font color="Green" title="E24">E1</font></span>: 
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">W</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">O</font></span><br/>

<a NAME="E2:29_2_1_1"/>
 not  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c6">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo7.html#T6" target="_self" onmouseover="rs('fintopo7/T6')" onmouseout="rh()">Th4</a></span>;<br/>
<a NAME="E3:29_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c6">x</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">v</font> = <font color="Maroon" title="c9">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ET</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_2_1_1"><span class="lab"><font color="Green" title="E24">E1</font></span></a></span>;<br/>
<a NAME="E5:29_2_1_1"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
( <font color="Maroon" title="c10">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ET</font> &amp; <font color="Maroon" title="c3">V</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c10">v</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_2_1_1"><span class="lab"><font color="Green" title="E24">E1</font></span></a>, <a class="txt" href="#E7:29_2_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a></span>;<br/>
<a NAME="E6:29_2_1_1"/>
 not  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c10">v</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo7.html#T6" target="_self" onmouseover="rs('fintopo7/T6')" onmouseout="rh()">Th4</a></span>;<br/>
<a NAME="E7:29_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">V</font> is    <a href="fintopo7.html#M1" title="FINTOPO7:mode.1">a_neighborhood</a> of <font color="Maroon" title="c10">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_2_1_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="ref" href="fintopo7.html#D5" target="_self" onmouseover="rs('fintopo7/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:29_2_1_1"/>
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">O</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:29_2_1"/>
( <font color="Maroon" title="c8">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c6">x</font> &amp;  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c6">x</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ET</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_2_1_2"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c6">x</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ET</font></span>
<div class="add"><a NAME="E1:29_2_1_2_1"/>
 not  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c6">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo7.html#T6" target="_self" onmouseover="rs('fintopo7/T6')" onmouseout="rh()">Th4</a></span>;<br/><span class="kw">hence </span><a NAME="E2:29_2_1_2_1"/>
<font color="Maroon" title="c8">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c6">x</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:29_2_1_2"/>
 <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c6">x</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ET</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo7.html#D2" target="_self" onmouseover="rs('fintopo7/D2')" onmouseout="rh()">Def2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:29_2_1"/>
<font color="Maroon" title="c5">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:29_2_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="card_fil.html#D1" onmouseover="rs('card_fil/D1')" onmouseout="rh()">CARD_FIL:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:29_2"/>
<font color="Maroon" title="c5">O</font> is   <a href="fintopo7.html#V1" title="FINTOPO7:attr.1">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">ET</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo7.html#D1" target="_self" onmouseover="rs('fintopo7/D1')" onmouseout="rh()">Def1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:29"/><span class="lab"><font color="Green" title="E20">A8</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">O</font>
 
;<br/>
<a NAME="E5:29"/>
<font color="Maroon" title="c5">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">V</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">O</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_3"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">O</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ET</font><span class="kw"> such that </span><br/><a NAME="E3:29_3"/><span class="lab"><font color="Green" title="E21">A9</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x0</font>
 <span class="kw">and </span><br/><a NAME="E4:29_3"/><span class="lab"><font color="Green" title="E22">A10</font></span>: 
<font color="Maroon" title="c3">V</font> is    <a href="fintopo7.html#M1" title="FINTOPO7:mode.1">a_neighborhood</a> of <font color="Maroon" title="c7">x0</font>
 ;<br/>
<a NAME="E5:29_3"/>
<font color="Maroon" title="c3">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c7">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_3"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="ref" href="fintopo7.html#D5" target="_self" onmouseover="rs('fintopo7/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:29_3"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_3"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="ref" href="fintopo7.html#D3" target="_self" onmouseover="rs('fintopo7/D3')" onmouseout="rh()">Def3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:29"/>
 ex <font color="Olive" title="b1">O</font> being   <a href="fintopo7.html#V1" title="FINTOPO7:attr.1">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">ET</font> st <br/>( <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">O</font> &amp; <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">V</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E4:29"><span class="lab"><font color="Green" title="E20">A8</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
