<?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"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><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></span>)</span><br/> for <font color="Olive" title="b2">B</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">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> holds <br/>( <font color="Olive" title="b1">A</font> is <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a> iff <font color="Olive" title="b2">B</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> )</span><br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><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></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">B</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="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> holds <br/>( <font color="Maroon" title="c2">A</font> is <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a> iff <font color="Olive" title="b1">B</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> )</span><br/><b>let </b><font color="Maroon" title="c3">B</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="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> implies ( <font color="Maroon" title="c2">A</font> is <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a> iff <font color="Maroon" title="c3">B</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> ) )</span><br/>





<b>assume </b><a NAME="E1:58"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a> iff <font color="Maroon" title="c3">B</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> )</span><br/>

<a NAME="E2:58"/><i><font color="Green" title="E39">A2</font></i>: 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><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></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="pre_topc.html#T5" title="PRE_TOPC:th.5">PRE_TOPC:5</a>, <a class="ref" href="yellow_1.html#T13" title="YELLOW_1:th.13">YELLOW_1:13</a></i>;<br/>
<a NAME="E3:58"/><i><font color="Green" title="E40">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><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></span>)</span> <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/>
<a NAME="E4:58"/><i><font color="Green" title="E41">A4</font></i>: 
<font color="Maroon" title="c3">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:58_1"/>
<font color="Maroon" title="c3">B</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="#E1:58"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="#E3:58"><i><font color="Green" title="E40">A3</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</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><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">B</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> implies <font color="Maroon" title="c2">A</font> is <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:58_2"/><i><font color="Green" title="E42">A5</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</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">B</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a></span><br/><a NAME="E2:58_2"/>
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/><font color="Maroon" title="c3">B</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="58_2_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 <font color="Maroon" title="c3">B</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><b>that </b><br/><a NAME="E1:58_2_1"/><i><font color="Green" title="E43">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> 
 <b>and </b><br/><a NAME="E2:58_2_1"/><i><font color="Green" title="E44">A7</font></i>: 
<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"> <font color="Maroon" title="c3">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">Q</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">q</font> = <font color="Maroon" title="c4">Q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><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></span>)</span> <b>by </b><i><a class="txt" href="#E3:58"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="#E2:58_2_1"><i><font color="Green" title="E44">A7</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="E4:58_2_1"/><i><font color="Green" title="E45">A8</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">q</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> <span class="p1">(<span class="default"><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></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:58"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="#E1:58_2"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="#E1:58_2_1"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="waybel12.html#D4" target="_self" title="WAYBEL12:def.4">Def4</a></i>;<br/>
<a NAME="E5:58_2_1"/>
<font color="Maroon" title="c3">B</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c4">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="#E4:58"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="#E2:58_2_1"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></i>;<br/>
<a NAME="E6:58_2_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">q</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="#E1:58"><i><font color="Green" title="E38">A1</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="E7:58_2_1"/><b>then </b>
<font color="Maroon" title="c3">B</font> <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="#E1:58"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="#E2:58"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="#E4:58_2_1"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="yellow_1.html#T9" title="YELLOW_1:th.9">YELLOW_1:9</a></i>;<br/>
<b>hence </b><a NAME="E8:58_2_1"/>
<font color="Maroon" title="c3">B</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><a NAME="E3:58_2"/><b>then </b>
<font color="Maroon" title="c3">B</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>;<br/><b>hence </b><a NAME="E4:58_2"/>
<font color="Maroon" title="c3">B</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 <b>by </b><i><a class="txt" href="#E4:58"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="tops_3.html#T36" title="TOPS_3:th.36">TOPS_3:36</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>assume </b><a NAME="E6:58"/>
<font color="Maroon" title="c3">B</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</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">A</font> is <a href="waybel12.html#V2" title="WAYBEL12:attr.2">dense</a></span><br/>

<a NAME="E7:58"/><b>then </b><i><font color="Green" title="E42">A9</font></i>: 
<font color="Maroon" title="c3">B</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#T33" title="TOPS_3:th.33">TOPS_3:33</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><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></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel12.html#D4" target="_self" title="WAYBEL12:def.4">WAYBEL12:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">v</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> <span class="p1">(<span class="default"><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></span>)</span> implies <font color="Maroon" title="c2">A</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">v</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> <span class="p1">(<span class="default"><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></span>)</span> )</span><br/>

<b>assume </b><a NAME="E8:58"/><i><font color="Green" title="E43">A10</font></i>: 
<font color="Maroon" title="c4">v</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> <span class="p1">(<span class="default"><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></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">A</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">v</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> <span class="p1">(<span class="default"><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></span>)</span></span><br/>

<a NAME="E9:58"/>
<font color="Maroon" title="c4">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="txt" href="#E3:58"><i><font color="Green" title="E40">A3</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">V</font> = <font color="Maroon" title="c4">v</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="E11:58"/><i><font color="Green" title="E44">A11</font></i>: 
<font color="Maroon" title="c5">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:58_3"/>
<font color="Maroon" title="c5">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="txt" href="#E3:58"><i><font color="Green" title="E40">A3</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</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>
<a NAME="E12:58"/><b>then </b>
<font color="Maroon" title="c3">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">V</font>
 
<b>by </b><i><a class="txt" href="#E2:58"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="#E7:58"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="#E8:58"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="tops_1.html#T80" title="TOPS_1:th.80">TOPS_1:80</a></i>;<br/>
<a NAME="E13:58"/><b>then </b><i><font color="Green" title="E45">A12</font></i>: 
<font color="Maroon" title="c3">B</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c5">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_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E14:58"/>
<font color="Maroon" title="c3">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:58_4"/>
<font color="Maroon" title="c3">B</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="#E1:58"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="#E3:58"><i><font color="Green" title="E40">A3</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</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>
<a NAME="E15:58"/><b>then </b>
<font color="Maroon" title="c3">B</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c5">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="#E11:58"><i><font color="Green" title="E44">A11</font></i></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></i>;<br/>
<a NAME="E16:58"/><b>then </b>
<font color="Maroon" title="c3">B</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c5">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/>
<b>hence </b><a NAME="E17:58"/>
<font color="Maroon" title="c2">A</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">v</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> <span class="p1">(<span class="default"><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></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:58"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="#E2:58"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="#E13:58"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="yellow_1.html#T9" title="YELLOW_1:th.9">YELLOW_1:9</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>
