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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">T</font> is <a href="yellow_8.html#V3" title="YELLOW_8:attr.3">sober</a> &amp; <font color="Maroon" title="c1">T</font> is <a href="waybel_3.html#V6" title="WAYBEL_3:attr.6">locally-compact</a> implies <font color="Maroon" title="c1">T</font> is <a href="yellow_8.html#V1" title="YELLOW_8:attr.1">Baire</a> )</span><br/>

<b>assume </b><a NAME="E1:61"/><i><font color="Green" title="E39">A1</font></i>: 
( <font color="Maroon" title="c1">T</font> is <a href="yellow_8.html#V3" title="YELLOW_8:attr.3">sober</a> &amp; <font color="Maroon" title="c1">T</font> is <a href="waybel_3.html#V6" title="WAYBEL_3:attr.6">locally-compact</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">T</font> is <a href="yellow_8.html#V1" title="YELLOW_8:attr.1">Baire</a></span><br/>

<b>let </b><font color="Maroon" title="c2">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel12.html#D6" target="_self" title="WAYBEL12:def.6">WAYBEL12:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a> &amp; ( for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/>( <font color="Olive" title="b1">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">S</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> ) ) implies  ex <font color="Olive" title="b1">I</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b1">I</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:61"/><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>
 <b>and </b><br/><a NAME="E3:61"/><i><font color="Green" title="E41">A3</font></i>: 
for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/>( <font color="Olive" title="b1">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">S</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">I</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b1">I</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c3">I</font> =  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c3">I</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c3">I</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )</span><br/>

<b>thus </b><a NAME="E5:61"/>
<font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">I</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a></span><br/>

<a NAME="E6:61"/><i><font color="Green" title="E42">A4</font></i>: 
( <font color="Maroon" title="c2">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="waybel12.html#V1" title="WAYBEL12:attr.1">dense</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:61_1"/>
<font color="Maroon" title="c2">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">F</font> is <a href="waybel12.html#V1" title="WAYBEL12:attr.1">dense</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> or <font color="Maroon" title="c4">X</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><br/>

<b>assume </b><a NAME="E1:61_1_1"/>
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">X</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/>

<b>hence </b><a NAME="E2:61_1_1"/>
<font color="Maroon" title="c4">X</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="#E3:61"><i><font color="Green" title="E41">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:61_1"/>
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">X</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 <b>by </b><i><a class="txt" href="#E3:61"><i><font color="Green" title="E41">A3</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel12.html#D2" target="_self" title="WAYBEL12:def.2">WAYBEL12:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_2"/>
( <font color="Maroon" title="c2">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1"><b>suppose </b></a><a NAME="E1:61_2_1"/><i><font color="Green" title="E43">A5</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">I</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a></span><br/><div class="add"><a NAME="E2:61_2_1"/>
for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/> <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">Q</font> )</span><br/>

<b>assume </b><a NAME="E1:61_2_1_1"/><i><font color="Green" title="E44">A6</font></i>: 
( <font color="Maroon" title="c4">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">Q</font></span><br/>

<b>then consider </b><font color="Maroon" title="c5">S</font> being  <a href="yellow_8.html#V2" title="YELLOW_8:attr.2">irreducible</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:61_2_1_1"/><i><font color="Green" title="E45">A7</font></i>: 
for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Maroon" title="c5">S</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c4">Q</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">V</font>
 <b>by </b><i><a class="txt" href="#E1:61"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E2:61"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="#E6:61"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="#E1:61_2_1"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="waybel12.html#T47" target="_self" title="WAYBEL12:th.47">Th47</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E5:61_2_1_1"/><i><font color="Green" title="E46">A8</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c5">S</font>
 <b>and </b><br/><a NAME="E6:61_2_1_1"/>
for <font color="Olive" title="b1">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">q</font> <a href="yellow_8.html#R1" title="YELLOW_8:pred.1">is_dense_point_of</a> <font color="Maroon" title="c5">S</font> holds <br/><font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font>
 <b>by </b><i><a class="txt" href="#E1:61"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="yellow_8.html#D6" title="YELLOW_8:def.6">YELLOW_8:def 6</a></i>;<br/>
<a NAME="E7:61_2_1_1"/>
<font color="Maroon" title="c5">S</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="ref" href="yellow_8.html#D4" title="YELLOW_8:def.4">YELLOW_8:def 4</a></i>;<br/>
<a NAME="E8:61_2_1_1"/><b>then </b><i><font color="Green" title="E47">A9</font></i>: 
<font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="#E5:61_2_1_1"><i><font color="Green" title="E46">A8</font></i></a>, <a class="ref" href="yellow_8.html#T25" title="YELLOW_8:th.25">YELLOW_8:25</a></i>;<br/>
<a NAME="E9:61_2_1_1"/><i><font color="Green" title="E48">A10</font></i>: 
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/>( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> implies ( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font> ) )</span><br/>

<b>assume </b><a NAME="E1:61_2_1_1_1"/><i><font color="Green" title="E49">A11</font></i>: 
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font> )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c8">Y1</font> = <font color="Maroon" title="c7">Y</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E3:61_2_1_1_1"/>
<font color="Maroon" title="c5">S</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c4">Q</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c8">Y1</font>
 
<b>by </b><i><a class="txt" href="#E3:61_2_1_1"><i><font color="Green" title="E45">A7</font></i></a>, <a class="txt" href="#E1:61_2_1_1_1"><i><font color="Green" title="E49">A11</font></i></a></i>;<br/>
<a NAME="E4:61_2_1_1_1"/><b>then </b><i><font color="Green" title="E50">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c4">Q</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c8">Y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_1_1"><b>now </b></a><div class="add"><a NAME="E1:61_2_1_1_1_1"/>
<font color="Maroon" title="c8">Y1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="#E3:61"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="#E1:61_2_1_1_1"><i><font color="Green" title="E49">A11</font></i></a></i>;<br/><a NAME="E2:61_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">Q</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c8">Y1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="#E1:61_2_1_1"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></i>;<br/><a NAME="E3:61_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E51">A13</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c8">Y1</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T30" title="TOPS_1:th.30">TOPS_1:30</a></i>;<br/><b>assume </b><a NAME="E4:61_2_1_1_1_1"/>
not <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c8">Y1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E5:61_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c8">Y1</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E6:61_2_1_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c8">Y1</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E7:61_2_1_1_1_1"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c8">Y1</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T49" title="PRE_TOPC:th.49">PRE_TOPC:49</a></i>;<br/><a NAME="E8:61_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c8">Y1</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="#E8:61_2_1_1"><i><font color="Green" title="E47">A9</font></i></a>, <a class="txt" href="#E3:61_2_1_1_1_1"><i><font color="Green" title="E51">A13</font></i></a>, <a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>;<br/><a NAME="E9:61_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">S</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Q</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c8">Y1</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/><a NAME="E10:61_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">S</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c8">Y1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E11:61_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:61_2_1_1_1"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:61_2_1_1_1"/>
( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E10:61_2_1_1"/><b>then </b>
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font>
 
;<br/>
<a NAME="E11:61_2_1_1"/><b>then </b><i><font color="Green" title="E49">A14</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:61_2_1_1"/><i><font color="Green" title="E50">A15</font></i>: 
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E1:61_2_1"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E14:61_2_1_1"/>
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font>
 
<b>by </b><i><a class="txt" href="#E9:61_2_1_1"><i><font color="Green" title="E48">A10</font></i></a>, <a class="txt" href="#E13:61_2_1_1"><i><font color="Green" title="E50">A15</font></i></a></i>;<br/>
<a NAME="E15:61_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c4">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="#E11:61_2_1_1"><i><font color="Green" title="E49">A14</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E16:61_2_1_1"/>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">Q</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:61_2_1"/>
<font color="Maroon" title="c3">I</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T80" title="TOPS_1:th.80">TOPS_1:80</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_2"><b>suppose </b></a><a NAME="E1:61_2_2"/>
<font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">I</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a></span><br/><div class="add"><a NAME="E2:61_2_2"/><b>then </b>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/><b>hence </b><a NAME="E3:61_2_2"/>
<font color="Maroon" title="c3">I</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T16" title="TOPS_3:th.16">TOPS_3:16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
