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

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">T</font> is  <a href="waybel_3.html#V6" title="WAYBEL_3:attr.6">locally-compact</a>  implies  for <font color="Olive" title="b1">D</font> being   <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st  not <font color="Olive" title="b1">D</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Olive" title="b1">D</font> is  <a href="waybel12.html#V1" title="WAYBEL12:attr.1">dense</a>  &amp; <font color="Olive" title="b1">D</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  holds <br/> for <font color="Olive" title="b2">O</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">T</font>  st <font color="Olive" title="b2">O</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b3">A</font> being   <a href="yellow_8.html#V3" title="YELLOW_8:attr.3">irreducible</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/> for <font color="Olive" title="b4">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="b4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> holds <br/><font color="Olive" title="b3">A</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Olive" title="b2">O</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b4">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:59"/>
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">D</font> being   <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st  not <font color="Olive" title="b1">D</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Olive" title="b1">D</font> is  <a href="waybel12.html#V1" title="WAYBEL12:attr.1">dense</a>  &amp; <font color="Olive" title="b1">D</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  holds <br/> for <font color="Olive" title="b2">O</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">T</font>  st <font color="Olive" title="b2">O</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b3">A</font> being   <a href="yellow_8.html#V3" title="YELLOW_8:attr.3">irreducible</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/> for <font color="Olive" title="b4">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="b4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> holds <br/><font color="Olive" title="b3">A</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Olive" title="b2">O</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b4">V</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">L</font> =  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> as   <a href="yellow_0.html#V3" title="YELLOW_0:attr.3">bounded</a>   <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#T42" title="WAYBEL_3:th.42">WAYBEL_3:42</a></span>;<br/>
<a NAME="E3:59"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">D</font> be   <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">D</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c3">D</font> is  <a href="waybel12.html#V1" title="WAYBEL12:attr.1">dense</a>  &amp; <font color="Maroon" title="c3">D</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  implies  for <font color="Olive" title="b1">O</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">T</font>  st <font color="Olive" title="b1">O</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b2">A</font> being   <a href="yellow_8.html#V3" title="YELLOW_8:attr.3">irreducible</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/> for <font color="Olive" title="b3">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="b3">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> holds <br/><font color="Olive" title="b2">A</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Olive" title="b1">O</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b3">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:59"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
(  not <font color="Maroon" title="c3">D</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c3">D</font> is  <a href="waybel12.html#V1" title="WAYBEL12:attr.1">dense</a>  &amp; <font color="Maroon" title="c3">D</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">O</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">T</font>  st <font color="Olive" title="b1">O</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b2">A</font> being   <a href="yellow_8.html#V3" title="YELLOW_8:attr.3">irreducible</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/> for <font color="Olive" title="b3">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="b3">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> holds <br/><font color="Olive" title="b2">A</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Olive" title="b1">O</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b3">V</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">I</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:59"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<font color="Maroon" title="c4">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">I</font> = <font color="Maroon" title="c4">I</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59"><span class="lab"><font color="Green" title="E41">A3</font></span></a></span>;<br/>
<a NAME="E8:59"/>
<font color="Maroon" title="c5">I</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E6:59"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">i</font> = <font color="Maroon" title="c5">I</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">D1</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> :  ex <font color="Olive" title="b2">d1</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="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b2">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> &amp; <font color="Olive" title="b1">d</font> is  <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a>  ) </span> } </span> ;<br/>
<a NAME="E10:59"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> :  ex <font color="Olive" title="b2">d1</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="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b2">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> &amp; <font color="Olive" title="b1">d</font> is  <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a>  ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> :  ex <font color="Olive" title="b2">d1</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="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b2">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> &amp; <font color="Olive" title="b1">d</font> is  <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a>  ) </span> } </span>  or <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_1"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> :  ex <font color="Olive" title="b2">d1</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="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b2">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> &amp; <font color="Olive" title="b1">d</font> is  <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a>  ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font></span><br/>

<a NAME="E2:59_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> st <br/>( <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> &amp;  ex <font color="Olive" title="b2">d1</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="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b2">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> &amp; <font color="Olive" title="b1">d</font> is  <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a>  ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:59_1"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">D1</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> :  ex <font color="Olive" title="b2">d1</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="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b2">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> &amp; <font color="Olive" title="b1">d</font> is  <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a>  ) </span> } </span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">L</font> ;<br/>
<a NAME="E12:59"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon" title="c8">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D1</font> or <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_2"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font></span><br/>

<a NAME="E2:59_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> st <br/>( <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> &amp;  ex <font color="Olive" title="b2">d1</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="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b2">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> &amp; <font color="Olive" title="b1">d</font> is  <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a>  ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:59_2"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 ; <a class="txt" 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="E13:59"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
<font color="Maroon" title="c8">D1</font> is  <a href="waybel12.html#V3" title="WAYBEL12:attr.3">dense</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel12.html#D5" target="_self" title="WAYBEL12:def.5">WAYBEL12:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D1</font> implies <font color="Maroon" title="c9">q</font> is  <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_3"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">q</font> is  <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a> </span><br/>

<a NAME="E2:59_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> st <br/>( <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> &amp;  ex <font color="Olive" title="b2">d1</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="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b2">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> &amp; <font color="Olive" title="b1">d</font> is  <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a>  ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:59_3"/>
<font color="Maroon" title="c9">q</font> is  <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a> 
 ; <a class="txt" 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">let </span><font color="Maroon" title="c9">O</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">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">O</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies  ex <font color="Olive" title="b1">A</font> being   <a href="yellow_8.html#V3" title="YELLOW_8:attr.3">irreducible</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/> for <font color="Olive" title="b2">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="b2">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> holds <br/><font color="Olive" title="b1">A</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c9">O</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E14:59"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
<font color="Maroon" title="c9">O</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A</font> being   <a href="yellow_8.html#V3" title="YELLOW_8:attr.3">irreducible</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/> for <font color="Olive" title="b2">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="b2">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> holds <br/><font color="Olive" title="b1">A</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c9">O</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">V</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">u</font> = <font color="Maroon" title="c9">O</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:59"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/>
<a NAME="E16:59"/>
( <font color="Maroon" title="c5">I</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c5">I</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E6:59"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="waybel12.html#D2" target="_self" title="WAYBEL12:def.2">Def2</a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></span>;<br/>
<a NAME="E17:59"/><span class="kw">then </span>
<font color="Maroon" title="c5">I</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_3.html#T36" title="TOPS_3:th.36">TOPS_3:36</a></span>;<br/>
<a NAME="E18:59"/><span class="kw">then </span>
<font color="Maroon" title="c6">i</font> is  <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel12.html#T42" target="_self" title="WAYBEL12:th.42">Th46</a></span>;<br/>
<a NAME="E19:59"/><span class="kw">then </span>
( <font color="Maroon" title="c10">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="pre_topc.html#T1" title="PRE_TOPC:th.1">PRE_TOPC:1</a>, <a class="ref" href="yellow_1.html#T13" title="YELLOW_1:th.13">YELLOW_1:13</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">p</font> being   <a href="waybel_6.html#NV3" title="WAYBEL_6:NV.3">irreducible</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font><span class="kw"> such that </span><br/><a NAME="E21:59"/><span class="lab"><font color="Green" title="E45">A7</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">and </span><br/><a NAME="E22:59"/><span class="lab"><font color="Green" title="E46">A8</font></span>: 
 not <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c8">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:59"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E13:59"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="waybel12.html#T39" target="_self" title="WAYBEL12:th.39">Th43</a></span>;<br/>
<a NAME="E23:59"/>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E39">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">P</font> = <font color="Maroon" title="c11">p</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">A</font> = <font color="Maroon" title="c12">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   <a href="yellow_8.html#V3" title="YELLOW_8:attr.3">irreducible</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:59"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="waybel12.html#T41" target="_self" title="WAYBEL12:th.41">Th45</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">A</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="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="c3">D</font> holds <br/><font color="Maroon" title="c13">A</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c9">O</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">V</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c14">V</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> implies <font color="Maroon" title="c13">A</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c9">O</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c14">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E26:59"/><span class="lab"><font color="Green" title="E47">A9</font></span>: 
<font color="Maroon" title="c14">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">A</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c9">O</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c14">V</font></span><br/>

<a NAME="E27:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A10</font></span>: 
<font color="Maroon" title="c14">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">v</font> = <font color="Maroon" title="c14">V</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/>
<a NAME="E29:59"/><span class="lab"><font color="Green" title="E49">A11</font></span>: 
 for <font color="Olive" title="b1">d1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b1">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D1</font> holds <br/> not <font color="Maroon" title="c10">u</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b1">d1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_4"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:59_4"/><span class="lab"><font color="Green" title="E50">A12</font></span>: 
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c16">d1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D1</font> implies  not <font color="Maroon" title="c10">u</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c16">d1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">p</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:59_4"/>
<font color="Maroon" title="c16">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c10">u</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c16">d1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">p</font></span><br/>

<a NAME="E3:59_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">u</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c16">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c8">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_4"><span class="lab"><font color="Green" title="E50">A12</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:59_4"/>
 not <font color="Maroon" title="c10">u</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c16">d1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:59"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="ref" href="waybel_0.html#D16" title="WAYBEL_0:def.16">WAYBEL_0:def 16</a></span>; <a class="txt" 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="E30:59"/>
<font color="Maroon" title="c14">V</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E26:59"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="waybel12.html#D2" target="_self" title="WAYBEL12:def.2">Def2</a></span>;<br/>
<a NAME="E31:59"/><span class="kw">then </span>
<font color="Maroon" title="c14">V</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:59"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="ref" href="tops_3.html#T36" title="TOPS_3:th.36">TOPS_3:36</a></span>;<br/>
<a NAME="E32:59"/><span class="kw">then </span>
<font color="Maroon" title="c15">v</font> is  <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel12.html#T42" target="_self" title="WAYBEL12:th.42">Th46</a></span>;<br/>
<a NAME="E33:59"/><span class="kw">then </span>
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:59"><span class="lab"><font color="Green" title="E47">A9</font></span></a></span>;<br/>
<a NAME="E34:59"/><span class="kw">then </span>
 not <font color="Maroon" title="c10">u</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c15">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:59"><span class="lab"><font color="Green" title="E49">A11</font></span></a></span>;<br/>
<a NAME="E35:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A13</font></span>: 
 not <font color="Maroon" title="c10">u</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c15">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></span>;<br/>
<a NAME="E36:59"/>
<font color="Maroon" title="c9">O</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c14">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:59"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="txt" href="#E27:59"><span class="lab"><font color="Green" title="E48">A10</font></span></a></span>;<br/>
<a NAME="E37:59"/><span class="kw">then </span>
<font color="Maroon" title="c10">u</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/>
<a NAME="E38:59"/><span class="kw">then </span>
 not <font color="Maroon" title="c9">O</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c14">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:59"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="ref" href="yellow_1.html#T9" title="YELLOW_1:th.9">YELLOW_1:9</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E40:59"/><span class="lab"><font color="Green" title="E51">A14</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">O</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c14">V</font>
 <span class="kw">and </span><br/><a NAME="E41:59"/><span class="lab"><font color="Green" title="E52">A15</font></span>: 
 not <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">x</font> = <font color="Maroon" title="c16">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:59"><span class="lab"><font color="Green" title="E51">A14</font></span></a></span>;<br/>
<a NAME="E43:59"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:59"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E44:59"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">O</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c14">V</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c13">A</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:59"><span class="lab"><font color="Green" title="E51">A14</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E45:59"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">A</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c9">O</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c14">V</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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
