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

<span class="kw">let </span><font color="Maroon" title="c1">ET</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="fintopo2.html#V1" title="FINTOPO2:attr.1">strict</a>   <a href="fintopo7.html#V2" title="FINTOPO7:attr.2">U_FMT_filter</a>   <a href="fintopo2.html#L1" title="FINTOPO2:struct.1">FMT_Space_Str</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   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> holds   <a href="fintopo7.html#K1" title="FINTOPO7:func.1">Neighborhood</a> <font color="Olive" title="b1">A</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><br/><span class="kw">let </span><font color="Maroon" title="c2">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">  <a href="fintopo7.html#K1" title="FINTOPO7:func.1">Neighborhood</a> <font color="Maroon" title="c2">A</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><br/>



<span class="kw">set </span><font color="Maroon" title="c3">S</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span> ;<br/>
<a NAME="E1:19"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  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="19_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:19_1"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  is   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</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="19_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:19_1_1"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
 not <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c4">Sq</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ET</font>;<br/>
<a NAME="E1:19_1_1_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ET</font> <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>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">Sq</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ET</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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/>
<a NAME="E3:19_1_1_1"/>
<font color="Maroon" title="c5">Sq</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:19_1_1_1_1"/>
 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="c2">A</font> holds <br/><font color="Maroon" title="c5">Sq</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="ref" href="fintopo7.html#T10" target="_self" onmouseover="rs('fintopo7/T10')" onmouseout="rh()">Th6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:19_1_1_1_1"/>
<font color="Maroon" title="c5">Sq</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">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>; <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="E4:19_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">Sq</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span> 
 
;<br/>
<span class="kw">hence </span><a NAME="E5:19_1_1_1"/>
 not <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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><span class="kw">end;</span></div>
<a NAME="E2:19_1_1"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</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="19_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:19_1_1_2"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</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="19_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">t</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="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  or <font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</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="E1:19_1_1_2_1"/>
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </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="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</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="c5">F0</font> being    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font><span class="kw"> such that </span><br/><a NAME="E3:19_1_1_2_1"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F0</font>
 ;<br/>
<span class="kw">thus </span><a NAME="E4:19_1_1_2_1"/>
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</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="#E3:19_1_1_2_1"><span class="lab"><font color="Green" title="E17">A3</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="E2:19_1_1_2"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:19_1_1"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  is   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</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="txt" href="#E1:19_1_1"><span class="lab"><font color="Green" title="E16">A2</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>
<a NAME="E2:19_1"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
 not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:19_1_2"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">F0</font> being    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font><span class="kw"> such that </span><br/><a NAME="E3:19_1_2"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F0</font>
 ;<br/>
<a NAME="E4:19_1_2"/>
 not <font color="Maroon" title="c2">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:19_1_2"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">a0</font> = <font color="Maroon" title="c5">a</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="#E6:19_1_2"><span class="lab"><font color="Green" title="E18">A6</font></span></a></span>;<br/>
<a NAME="E8:19_1_2"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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">a0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_1_2"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E6:19_1_2"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="ref" href="fintopo7.html#D6" target="_self" onmouseover="rs('fintopo7/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E9:19_1_2"/>
 <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c6">a0</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>;<br/>
<span class="kw">hence </span><a NAME="E10:19_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:19_1_2"><span class="lab"><font color="Green" title="E19">A7</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>
<a NAME="E3:19_1"/>
 for <font color="Olive" title="b1">Y1</font>, <font color="Olive" title="b2">Y2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">ET</font> holds <br/> ( ( <font color="Olive" title="b1">Y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b3">F</font> where <font color="Olive" title="b3">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  &amp; <font color="Olive" title="b2">Y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b3">F</font> where <font color="Olive" title="b3">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  implies <font color="Olive" title="b1">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">Y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b3">F</font> where <font color="Olive" title="b3">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  ) &amp; ( <font color="Olive" title="b1">Y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b3">F</font> where <font color="Olive" title="b3">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  &amp; <font color="Olive" title="b1">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y2</font> implies <font color="Olive" title="b2">Y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b3">F</font> where <font color="Olive" title="b3">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">Y1</font>, <font color="Maroon" title="c5">Y2</font> be   <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="c4">Y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  &amp; <font color="Maroon" title="c5">Y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  implies <font color="Maroon" title="c4">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  ) &amp; ( <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  &amp; <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</font> implies <font color="Maroon" title="c5">Y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  ) )</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1_3_1"><span class="kw">now </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="c4">Y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  &amp; <font color="Maroon" title="c5">Y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  implies <font color="Maroon" title="c4">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  )</span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:19_1_3_1"/><span class="lab"><font color="Green" title="E17">A8</font></span>: 
<font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E2:19_1_3_1"/><span class="lab"><font color="Green" title="E18">A9</font></span>: 
<font color="Maroon" title="c5">Y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </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="c4">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span> </span><br/><span class="kw">consider </span><font color="Maroon" title="c6">F1</font> being    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font><span class="kw"> such that </span><br/><a NAME="E4:19_1_3_1"/><span class="lab"><font color="Green" title="E19">A10</font></span>: 
<font color="Maroon" title="c4">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_3_1"><span class="lab"><font color="Green" title="E17">A8</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">F2</font> being    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font><span class="kw"> such that </span><br/><a NAME="E6:19_1_3_1"/><span class="lab"><font color="Green" title="E20">A11</font></span>: 
<font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_3_1"><span class="lab"><font color="Green" title="E18">A9</font></span></a></span>;<br/><a NAME="E7:19_1_3_1"/><span class="lab"><font color="Green" title="E21">A12</font></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="c2">A</font> holds <br/><font color="Maroon" title="c4">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y2</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="19_1_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">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="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> implies <font color="Maroon" title="c4">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y2</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="c8">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:19_1_3_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <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"> <font color="Maroon" title="c4">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y2</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="c8">x</font></span><br/>

<a NAME="E2:19_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A13</font></span>: 
( <font color="Maroon" title="c4">Y1</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="c8">x</font> &amp; <font color="Maroon" title="c5">Y2</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="c8">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1_3_1"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E6:19_1_3_1"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="ref" href="fintopo7.html#D6" target="_self" onmouseover="rs('fintopo7/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E3:19_1_3_1_1"/>
 <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c8">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>;<br/>
<span class="kw">hence </span><a NAME="E4:19_1_3_1_1"/>
<font color="Maroon" title="c4">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y2</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="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_3_1_1"><span class="lab"><font color="Green" title="E22">A13</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><a NAME="E8:19_1_3_1"/>
<font color="Maroon" title="c4">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y2</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:19_1_3_1"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="ref" href="fintopo7.html#D6" target="_self" onmouseover="rs('fintopo7/D6')" onmouseout="rh()">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E9:19_1_3_1"/>
<font color="Maroon" title="c4">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </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"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:19_1_3"/>
( <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  &amp; <font color="Maroon" title="c5">Y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  implies <font color="Maroon" title="c4">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </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="c4">Y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  &amp; <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</font> implies <font color="Maroon" title="c5">Y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  )</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1_3_2"><span class="kw">now </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="c4">Y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  &amp; <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</font> &amp; <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  &amp; <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</font> implies <font color="Maroon" title="c5">Y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  )</span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:19_1_3_2"/><span class="lab"><font color="Green" title="E17">A14</font></span>: 
<font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E2:19_1_3_2"/><span class="lab"><font color="Green" title="E18">A15</font></span>: 
<font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</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="c4">Y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  &amp; <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</font> implies <font color="Maroon" title="c5">Y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  )</span><br/><span class="kw">consider </span><font color="Maroon" title="c6">y1</font> being    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font><span class="kw"> such that </span><br/><a NAME="E4:19_1_3_2"/><span class="lab"><font color="Green" title="E19">A16</font></span>: 
<font color="Maroon" title="c6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_3_2"><span class="lab"><font color="Green" title="E17">A14</font></span></a></span>;<br/><a NAME="E5:19_1_3_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="c2">A</font> holds <br/><font color="Maroon" title="c5">Y2</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="19_1_3_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> implies <font color="Maroon" title="c5">Y2</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">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:19_1_3_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <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"> <font color="Maroon" title="c5">Y2</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">x</font></span><br/>

<a NAME="E2:19_1_3_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A17</font></span>: 
<font color="Maroon" title="c4">Y1</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">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1_3_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="ref" href="fintopo7.html#D6" target="_self" onmouseover="rs('fintopo7/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E3:19_1_3_2_1"/>
 <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c7">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>;<br/>
<span class="kw">hence </span><a NAME="E4:19_1_3_2_1"/>
<font color="Maroon" title="c5">Y2</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">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_3_2_1"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E2:19_1_3_2"><span class="lab"><font color="Green" title="E18">A15</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><a NAME="E6:19_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">Y2</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">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/><span class="kw">hence </span><a NAME="E7:19_1_3_2"/>
( <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  &amp; <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</font> implies <font color="Maroon" title="c5">Y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </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"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:19_1_3"/>
( <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  &amp; <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</font> implies <font color="Maroon" title="c5">Y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:19_1"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is    <a href="fintopo7.html#M2" title="FINTOPO7:mode.2">a_neighborhood</a> of <font color="Maroon" title="c2">A</font> : verum </span> } </span>  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="txt" href="#E1:19_1"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E2:19_1"><span class="lab"><font color="Green" title="E16">A4</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="E2:19"/>
 <a href="fintopo7.html#K1" title="FINTOPO7:func.1">Neighborhood</a> <font color="Maroon" title="c2">A</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>
 ; <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>
