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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>assume </b><a NAME="E1:71"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="urysohn1.html#V1">being_T1</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>] means <font color="Maroon">a<sub>1</sub></font> is <a href="topgen_1.html#V2" target="_self">dense-in-itself</a>;<br/>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:71"/><i><font color="Green">E42</font></i>: 
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> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E4:71"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="topgen_1.html#V4" target="_self">dense-in-itself</a>
 
<b>by </b><i><a class="txt" href="topgen_1.html#E3:71"><i><font color="Green">E42</font></i></a>, <a class="ref" href="topgen_1.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E5:71"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</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#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E6:71"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="topgen_1.html#V2" target="_self">dense-in-itself</a>
 
<b>by </b><i><a class="txt" href="topgen_1.html#E1:71"><i><font color="Green">E41</font></i></a>, <a class="ref" href="topgen_1.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E7:71"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_1.html#E3:71"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E8:71"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E9:71"/>
 <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="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E10:71"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_1.html#E8:71"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E11:71"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> is <a href="topgen_1.html#V5" target="_self">perfect</a>
 
<b>by </b><i><a class="txt" href="topgen_1.html#E5:71"><i><font color="Green">E43</font></i></a>, <a class="ref" href="topgen_1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E12:71"/>
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> holds <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  or  not <font color="Olive">b<sub>1</sub></font> is <a href="topgen_1.html#V2" target="_self">dense-in-itself</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>4</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="E1:71_1"/><i><font color="Green">E46</font></i>: 
( not <font color="Maroon">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="topgen_1.html#V2" target="_self">dense-in-itself</a> )
 ;<br/>

<a NAME="E2:71_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_1.html#E1:71_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<a NAME="E3:71_1"/>
<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="txt" href="topgen_1.html#E1:71_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="topgen_1.html#E3:71"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E4:71_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E5:71_1"/>
contradiction
 <b>by </b><i><a class="txt" href="topgen_1.html#E1:71_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="topgen_1.html#E2:71_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_1.html#T69">XBOOLE_1:69</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:71"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="topgen_1.html#V6" target="_self">scattered</a>
 
<b>by </b><i><a class="ref" href="topgen_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E14:71"/><i><font color="Green">E47</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T18">PRE_TOPC:18</a></i>;<br/>
<a NAME="E15:71"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<b>hence </b><a NAME="E16:71"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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="subset_1.html#K4">\/</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="topgen_1.html#V5" target="_self">perfect</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="topgen_1.html#V6" target="_self">scattered</a> )
 <b>by </b><i><a class="txt" href="topgen_1.html#E11:71"><i><font color="Green">E45</font></i></a>, <a class="txt" href="topgen_1.html#E13:71"><i><font color="Green">E46</font></i></a>, <a class="txt" href="topgen_1.html#E14:71"><i><font color="Green">E47</font></i></a></i>;<br/>


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