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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#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:52"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="topgen_1.html#V4" target="_self">dense-in-itself</a>
 ;<br/>

<b>thus </b><a NAME="E2:52"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" target="_self">Der</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:52_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<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#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="topgen_1.html#E3:52_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E5:52_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="topgen_1.html#V2" target="_self">dense-in-itself</a>
 
<b>by </b><i><a class="ref" href="topgen_1.html#D9" target="_self">Def9</a>, <a class="txt" href="topgen_1.html#E1:52"><i><font color="Green">E33</font></i></a>, <a class="txt" href="topgen_1.html#E3:52_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E6:52_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="topgen_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E7:52_1"/>
 <a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="topgen_1.html#K3" target="_self">Der</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="topgen_1.html#D6" target="_self">Def6</a>, <a class="txt" href="topgen_1.html#E3:52_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:52_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" target="_self">Der</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="topgen_1.html#E6:52_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a>, <a class="txt" href="topgen_1.html#E3:52_1"><i><font color="Green">E34</font></i></a></i>;<br/>


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


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