<?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">cF</font> being    <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><br/>  for <font color="Olive" title="b2">cS</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b1">cF</font><br/>  for <font color="Olive" title="b3">A</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">ET</font>  st <font color="Olive" title="b1">cF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fintopo7.html#K1" title="FINTOPO7:func.1">Neighborhood</a> <font color="Olive" title="b3">A</font> &amp; <font color="Olive" title="b2">cS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fintopo7.html#K5" title="FINTOPO7:func.5">OpenNeighborhoods</a> <font color="Olive" title="b3">A</font> holds <br/><font color="Olive" title="b2">cS</font> is  <a href="cardfil2.html#V3" title="CARDFIL2:attr.3">filter_basis</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">cF</font> be    <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>; <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">cS</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">cF</font><br/>  for <font color="Olive" title="b2">A</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">ET</font>  st <font color="Maroon" title="c2">cF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fintopo7.html#K1" title="FINTOPO7:func.1">Neighborhood</a> <font color="Olive" title="b2">A</font> &amp; <font color="Olive" title="b1">cS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fintopo7.html#K5" title="FINTOPO7:func.5">OpenNeighborhoods</a> <font color="Olive" title="b2">A</font> holds <br/><font color="Olive" title="b1">cS</font> is  <a href="cardfil2.html#V3" title="CARDFIL2:attr.3">filter_basis</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">cS</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">cF</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">A</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">ET</font>  st <font color="Maroon" title="c2">cF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fintopo7.html#K1" title="FINTOPO7:func.1">Neighborhood</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c3">cS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fintopo7.html#K5" title="FINTOPO7:func.5">OpenNeighborhoods</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Maroon" title="c3">cS</font> is  <a href="cardfil2.html#V3" title="CARDFIL2:attr.3">filter_basis</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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"> ( <font color="Maroon" title="c2">cF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fintopo7.html#K1" title="FINTOPO7:func.1">Neighborhood</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c3">cS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fintopo7.html#K5" title="FINTOPO7:func.5">OpenNeighborhoods</a> <font color="Maroon" title="c4">A</font> implies <font color="Maroon" title="c3">cS</font> is  <a href="cardfil2.html#V3" title="CARDFIL2:attr.3">filter_basis</a>  )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c2">cF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fintopo7.html#K1" title="FINTOPO7:func.1">Neighborhood</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E2:32"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<font color="Maroon" title="c3">cS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fintopo7.html#K5" title="FINTOPO7:func.5">OpenNeighborhoods</a> <font color="Maroon" title="c4">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"> <font color="Maroon" title="c3">cS</font> is  <a href="cardfil2.html#V3" title="CARDFIL2:attr.3">filter_basis</a> </span><br/>

<a NAME="E3:32"/>
 for <font color="Olive" title="b1">f</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">cF</font>  ex <font color="Olive" title="b2">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">cS</font> st <font color="Olive" title="b2">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">f</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">f</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">cF</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">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">cS</font> st <font color="Olive" title="b1">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">f</font></span><br/>

<a NAME="E1:32_1"/>
<font color="Maroon" title="c5">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">N</font> where <font color="Olive" title="b1">N</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c4">A</font> : verum </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E20">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">N</font> being    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c4">A</font><span class="kw"> such that </span><br/><a NAME="E3:32_1"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">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><span class="kw"> such that </span><br/><a NAME="E5:32_1"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
( <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">O</font> &amp; <font color="Maroon" title="c7">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">N</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo7.html#T16" target="_self" onmouseover="rs('fintopo7/T16')" onmouseout="rh()">Th11</a></span>;<br/>
<a NAME="E6:32_1"/>
<font color="Maroon" title="c7">O</font> is  <a href="fintopo7.html#V1" title="FINTOPO7:attr.1">open</a> 
 
;<br/>
<a NAME="E7:32_1"/><span class="kw">then </span>
 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="c4">A</font> holds <br/><font color="Maroon" title="c7">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a></span>;<br/>
<a NAME="E8:32_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">O</font> is   <a href="fintopo7.html#V1" title="FINTOPO7:attr.1">open</a>   <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo7.html#D6" target="_self" onmouseover="rs('fintopo7/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E9:32_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">N</font> where <font color="Olive" title="b1">N</font> is   <a href="fintopo7.html#V1" title="FINTOPO7:attr.1">open</a>   <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c4">A</font> : verum </span> } </span> 
 
;<br/>
<span class="kw">hence </span><a NAME="E10:32_1"/>
 ex <font color="Olive" title="b1">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">cS</font> st <font color="Olive" title="b1">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:32_1"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E5:32_1"><span class="lab"><font color="Green" title="E23">A4</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="E4:32"/>
<font color="Maroon" title="c3">cS</font> is  <a href="cardfil2.html#V3" title="CARDFIL2:attr.3">filter_basis</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"> verum</span><br/>


</div>
