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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:68"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:68"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T5">PRE_TOPC:5</a></i>;<br/>
<a NAME="E3:68"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="yellow_1.html#T13">YELLOW_1:13</a></i>;<br/>
<a NAME="E4:68"/><i><font color="Green">E44</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E5:68"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:68_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel12.html#E1:68"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel12.html#E4:68"><i><font color="Green">E44</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a><br/>


</div><b>end;</b></div>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:68_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel12.html#V2" target="_self">dense</a>
 ;<br/><a NAME="E2:68_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:68_2_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:68_2_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel12.html#E4:68"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel12.html#E2:68_2_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E4:68_2_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel12.html#E3:68"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel12.html#E1:68_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel12.html#E1:68_2_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="waybel12.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E5:68_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="waybel12.html#E5:68"><i><font color="Green">E45</font></i></a>, <a class="txt" href="waybel12.html#E2:68_2_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>
<a NAME="E6:68_2_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E1:68"><i><font color="Green">E42</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E7:68_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="waybel12.html#E1:68"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel12.html#E3:68"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel12.html#E4:68_2_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="yellow_1.html#T9">YELLOW_1:9</a></i>;<br/>
<b>hence </b><a NAME="E8:68_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:68_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V1">dense</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T80">TOPS_1:80</a></i>;<br/><b>hence </b><a NAME="E4:68_2"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_3.html#V1">everywhere_dense</a>
 <b>by </b><i><a class="txt" href="waybel12.html#E5:68"><i><font color="Green">E45</font></i></a>, <a class="ref" href="tops_3.html#T36">TOPS_3:36</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E7:68"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_3.html#V1">everywhere_dense</a>
 ;<br/>

<a NAME="E8:68"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V1">dense</a>
 
<b>by </b><i><a class="ref" href="tops_3.html#T33">TOPS_3:33</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel12.html#D4" target="_self">WAYBEL12:def 4</a><br/>

<b>assume </b><a NAME="E9:68"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E10:68"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E4:68"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E12:68"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:68_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel12.html#E4:68"><i><font color="Green">E44</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a><br/>


</div><b>end;</b></div>
<a NAME="E13:68"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E3:68"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel12.html#E8:68"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel12.html#E9:68"><i><font color="Green">E47</font></i></a>, <a class="ref" href="tops_1.html#T80">TOPS_1:80</a></i>;<br/>
<a NAME="E14:68"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E15:68"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:68_4"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel12.html#E1:68"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel12.html#E4:68"><i><font color="Green">E44</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a><br/>


</div><b>end;</b></div>
<a NAME="E16:68"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="waybel12.html#E12:68"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>
<a NAME="E17:68"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E18:68"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel12.html#E1:68"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel12.html#E3:68"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel12.html#E14:68"><i><font color="Green">E49</font></i></a>, <a class="ref" href="yellow_1.html#T9">YELLOW_1:9</a></i>;<br/>


</div>
