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<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#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:43"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a> implies ex <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 <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V1">dense</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 ;<br/>

<b>take </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>thus </b><a NAME="E2:43_1"/>
(  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V1">dense</a> )
 <b>by </b><i><a class="txt" href="tops_3.html#E1:43_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tops_3.html#T35" target="_self">Th35</a>, <a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>


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

<b>given </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E2:43"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <b>and </b><br/><a NAME="E3:43"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V1">dense</a>
 ;<br/>

<a NAME="E4:43"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="tops_3.html#E2:43"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tops_1.html#T56">TOPS_1:56</a></i>;<br/>
<a NAME="E5:43"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V1">dense</a>
 
<b>by </b><i><a class="txt" href="tops_3.html#E3:43"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tops_1.html#T79">TOPS_1:79</a></i>;<br/>
<b>hence </b><a NAME="E6:43"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T35" target="_self">Th35</a></i>;<br/>


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