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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:69"/>
<font color="Maroon" title="c1">T</font> is <a href="waybel_3.html#V6" title="WAYBEL_3:attr.6">locally-compact</a>
 ;<br/>

<b>then reconsider </b><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> <b>by </b><i><a class="ref" href="waybel_3.html#T43" title="WAYBEL_3:th.43">WAYBEL_3:43</a></i>;<br/>
<b>let </b><font color="Maroon" title="c3">D</font> be  <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><a NAME="E3:69"/><i><font color="Green" title="E43">A1</font></i>: 
( 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> )
 ;<br/>

<b>let </b><font color="Maroon" title="c4">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>;<br/>

<b>assume </b><a NAME="E4:69"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c4">O</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E5:69"/><i><font color="Green" title="E45">A3</font></i>: 
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>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">u</font> = <font color="Maroon" title="c4">O</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> <b>by </b><i><a class="txt" href="waybel12.html#E4:69"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E7:69"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T5" title="PRE_TOPC:th.5">PRE_TOPC:5</a></i>;<br/>
<a NAME="E8:69"/><b>then </b><i><font color="Green" title="E46">A4</font></i>: 
<font color="Maroon" title="c5">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>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T13" title="YELLOW_1:th.13">YELLOW_1:13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">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="E9:69"/>
<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="69_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:69_1"/>
<font color="Maroon" title="c7">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> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E3:69_1"/><i><font color="Green" title="E47">A5</font></i>: 
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d</font>
 <b>and </b><br/><a NAME="E4:69_1"/>
 ex <font color="Olive" title="b1">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="b1">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="b1">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c8">d</font> is <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a> )
 ;<br/>
<b>thus </b><a NAME="E5:69_1"/>
<font color="Maroon" title="c7">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>
 <b>by </b><i><a class="txt" href="waybel12.html#E3:69_1"><i><font color="Green" title="E47">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><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>  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="E11:69"/>
<font color="Maroon" title="c7">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="69_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:69_2"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E3:69_2"/><i><font color="Green" title="E47">A6</font></i>: 
( <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d</font> &amp;  ex <font color="Olive" title="b1">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="b1">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="b1">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d</font> &amp; <font color="Maroon" title="c9">d</font> is <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a> ) )
 ;<br/>
<b>thus </b><a NAME="E4:69_2"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E3:69_2"><i><font color="Green" title="E47">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:69"/><b>then </b><i><font color="Green" title="E47">A7</font></i>: 
<font color="Maroon" title="c7">D1</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 
<b>by </b><i><a class="ref" href="card_4.html#T46" title="CARD_4:th.46">CARD_4:46</a></i>;<br/>
<a NAME="E13:69"/><i><font color="Green" title="E48">A8</font></i>: 
<font color="Maroon" title="c7">D1</font> is <a href="waybel12.html#V3" title="WAYBEL12:attr.3">dense</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel12.html#D5" target="_self" title="WAYBEL12:def.5">WAYBEL12:def 5</a><br/>

<b>assume </b><a NAME="E1:69_3"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E3:69_3"/><i><font color="Green" title="E49">A9</font></i>: 
( <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d</font> &amp;  ex <font color="Olive" title="b1">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="b1">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="b1">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d</font> &amp; <font color="Maroon" title="c9">d</font> is <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a> ) )
 ;<br/>
<b>thus </b><a NAME="E4:69_3"/>
<font color="Maroon" title="c8">q</font> is <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a>
 <b>by </b><i><a class="txt" href="waybel12.html#E3:69_3"><i><font color="Green" title="E49">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c8">I</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:69"/><i><font color="Green" title="E49">A10</font></i>: 
<font color="Maroon" title="c8">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E3:69"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">I</font> = <font color="Maroon" title="c8">I</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="waybel12.html#E15:69"><i><font color="Green" title="E49">A10</font></i></a></i>;<br/>
<a NAME="E17:69"/>
<font color="Maroon" title="c9">I</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel12.html#E3:69"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="waybel12.html#E15:69"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">i</font> = <font color="Maroon" title="c9">I</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> <b>by </b><i><a class="txt" href="waybel12.html#E5:69"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E19:69"/>
( <font color="Maroon" title="c9">I</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c9">I</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )
 
<b>by </b><i><a class="txt" href="waybel12.html#E3:69"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="waybel12.html#E15:69"><i><font color="Green" title="E49">A10</font></i></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></i>;<br/>
<a NAME="E20:69"/><b>then </b>
<font color="Maroon" title="c9">I</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 
<b>by </b><i><a class="ref" href="tops_3.html#T36" title="TOPS_3:th.36">TOPS_3:36</a></i>;<br/>
<a NAME="E21:69"/><b>then </b>
<font color="Maroon" title="c10">i</font> is <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a>
 
<b>by </b><i><a class="ref" href="waybel12.html#T46" target="_self" title="WAYBEL12:th.46">Th46</a></i>;<br/>
<a NAME="E22:69"/><b>then </b>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D1</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E15:69"><i><font color="Green" title="E49">A10</font></i></a></i>;<br/>
<b>then consider </b><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><b> such that </b><br/><a NAME="E24:69"/><i><font color="Green" title="E50">A11</font></i>: 
<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>
 <b>and </b><br/><a NAME="E25:69"/><i><font color="Green" title="E51">A12</font></i>: 
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="c5">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="c7">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel12.html#E8:69"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="waybel12.html#E12:69"><i><font color="Green" title="E47">A7</font></i></a>, <a class="txt" href="waybel12.html#E13:69"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="waybel12.html#T43" target="_self" title="WAYBEL12:th.43">Th43</a></i>;<br/>
<a NAME="E26:69"/>
<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>
 
<b>by </b><i><a class="txt" href="waybel12.html#E5:69"><i><font color="Green" title="E45">A3</font></i></a></i>;<br/>
<b>then reconsider </b><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/>
<b>reconsider </b><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#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>by </b><i><a class="txt" href="waybel12.html#E24:69"><i><font color="Green" title="E50">A11</font></i></a>, <a class="ref" href="waybel12.html#T45" target="_self" title="WAYBEL12:th.45">Th45</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">A</font>
;<br/>

<b>let </b><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>;<br/>

<b>assume </b><a NAME="E29:69"/><i><font color="Green" title="E52">A13</font></i>: 
<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>
 ;<br/>

<a NAME="E30:69"/><b>then </b><i><font color="Green" title="E53">A14</font></i>: 
<font color="Maroon" title="c14">V</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 
<b>by </b><i><a class="txt" href="waybel12.html#E3:69"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="waybel12.html#D2" target="_self" title="WAYBEL12:def.2">Def2</a></i>;<br/>
<a NAME="E31:69"/><i><font color="Green" title="E54">A15</font></i>: 
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="c7">D1</font> holds <br/> not <font color="Maroon" title="c5">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="69_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">d1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>;<br/>

<b>assume </b><a NAME="E1:69_4"/><i><font color="Green" title="E55">A16</font></i>: 
<font color="Maroon" title="c15">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D1</font>
 ;<br/>

<a NAME="E2:69_4"/>
<font color="Maroon" title="c5">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="c5">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:69_4"/><b>then </b>
<font color="Maroon" title="c5">u</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c15">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="c5">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="c7">D1</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E1:69_4"><i><font color="Green" title="E55">A16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:69_4"/>
not <font color="Maroon" title="c5">u</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c15">d1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">p</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E25:69"><i><font color="Green" title="E51">A12</font></i></a>, <a class="ref" href="waybel_0.html#D16" title="WAYBEL_0:def.16">WAYBEL_0:def 16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:69"/><i><font color="Green" title="E55">A17</font></i>: 
<font color="Maroon" title="c14">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel12.html#E3:69"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="waybel12.html#E29:69"><i><font color="Green" title="E52">A13</font></i></a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></i>;<br/>
<b>then reconsider </b><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> <b>by </b><i><a class="txt" href="waybel12.html#E5:69"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E34:69"/>
<font color="Maroon" title="c4">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>
 
<b>by </b><i><a class="txt" href="waybel12.html#E4:69"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="waybel12.html#E32:69"><i><font color="Green" title="E55">A17</font></i></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></i>;<br/>
<a NAME="E35:69"/><b>then </b><i><font color="Green" title="E56">A18</font></i>: 
<font color="Maroon" title="c5">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>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E36:69"/>
<font color="Maroon" title="c14">V</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 
<b>by </b><i><a class="txt" href="waybel12.html#E30:69"><i><font color="Green" title="E53">A14</font></i></a>, <a class="txt" href="waybel12.html#E32:69"><i><font color="Green" title="E55">A17</font></i></a>, <a class="ref" href="tops_3.html#T36" title="TOPS_3:th.36">TOPS_3:36</a></i>;<br/>
<a NAME="E37:69"/><b>then </b>
<font color="Maroon" title="c15">v</font> is <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a>
 
<b>by </b><i><a class="ref" href="waybel12.html#T46" target="_self" title="WAYBEL12:th.46">Th46</a></i>;<br/>
<a NAME="E38:69"/><b>then </b>
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D1</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E29:69"><i><font color="Green" title="E52">A13</font></i></a></i>;<br/>
<a NAME="E39:69"/><b>then </b>
not <font color="Maroon" title="c5">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>
 
<b>by </b><i><a class="txt" href="waybel12.html#E31:69"><i><font color="Green" title="E54">A15</font></i></a></i>;<br/>
<a NAME="E40:69"/><b>then </b>
not <font color="Maroon" title="c5">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>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>
<a NAME="E41:69"/><b>then </b>
not <font color="Maroon" title="c4">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>
 
<b>by </b><i><a class="txt" href="waybel12.html#E35:69"><i><font color="Green" title="E56">A18</font></i></a>, <a class="ref" href="yellow_1.html#T9" title="YELLOW_1:th.9">YELLOW_1:9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E43:69"/><i><font color="Green" title="E57">A19</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">O</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c14">V</font>
 <b>and </b><br/><a NAME="E44:69"/><i><font color="Green" title="E58">A20</font></i>: 
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> 
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><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> <b>by </b><i><a class="txt" href="waybel12.html#E43:69"><i><font color="Green" title="E57">A19</font></i></a></i>;<br/>
<a NAME="E46:69"/>
<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>
 
<b>by </b><i><a class="txt" href="waybel12.html#E44:69"><i><font color="Green" title="E58">A20</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E47:69"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">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> 
 
<b>by </b><i><a class="txt" href="waybel12.html#E43:69"><i><font color="Green" title="E57">A19</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E48:69"/>
<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="c4">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> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/>


</div>
