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<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>;<br/>

<b>assume </b><a NAME="E1:71"/><i><font color="Green" title="E39">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="urysohn1.html#V1" title="URYSOHN1:attr.1">being_T1</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>] <b>means </b>$1 is <a href="topgen_1.html#V2" title="TOPGEN_1:attr.2">dense-in-itself</a>;<br/>
<b>consider </b><font color="Maroon" title="c2">CC</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:71"/><i><font color="Green" title="E40">A2</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">CC</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3" title="SUBSET_1:sch.3">SUBSET_1:sch 3</a>();<br/></i>
<b>set </b><font color="Maroon" title="c3">C</font> =  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">CC</font>;<br/>
<a NAME="E4:71"/><i><font color="Green" title="E41">A3</font></i>: 
<font color="Maroon" title="c2">CC</font> is <a href="topgen_1.html#V4" title="TOPGEN_1:attr.4">dense-in-itself</a>
 
<b>by </b><i><a class="txt" href="topgen_1.html#E3:71"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="topgen_1.html#D9" target="_self" title="TOPGEN_1:def.9">Def9</a></i>;<br/>
<a NAME="E5:71"/><b>then </b><i><font color="Green" title="E42">A4</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">CC</font> is <a href="topgen_1.html#V2" title="TOPGEN_1:attr.2">dense-in-itself</a>
 
<b>by </b><i><a class="ref" href="topgen_1.html#T40" target="_self" title="TOPGEN_1:th.40">Th40</a></i>;<br/>
<a NAME="E6:71"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">CC</font></span>)</span> is <a href="topgen_1.html#V2" title="TOPGEN_1:attr.2">dense-in-itself</a>
 
<b>by </b><i><a class="txt" href="topgen_1.html#E1:71"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="topgen_1.html#E4:71"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="topgen_1.html#T38" target="_self" title="TOPGEN_1:th.38">Th38</a>, <a class="ref" href="topgen_1.html#T40" target="_self" title="TOPGEN_1:th.40">Th40</a></i>;<br/>
<a NAME="E7:71"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">CC</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">CC</font>
 
<b>by </b><i><a class="txt" href="topgen_1.html#E3:71"><i><font color="Green" title="E40">A2</font></i></a></i>;<br/>
<a NAME="E8:71"/><b>then </b><i><font color="Green" title="E43">A5</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">CC</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">CC</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E9:71"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">CC</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">CC</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E10:71"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">CC</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">CC</font>
 
<b>by </b><i><a class="txt" href="topgen_1.html#E8:71"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E11:71"/><b>then </b><i><font color="Green" title="E44">A6</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">CC</font> is <a href="topgen_1.html#V5" title="TOPGEN_1:attr.5">perfect</a>
 
<b>by </b><i><a class="txt" href="topgen_1.html#E5:71"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="topgen_1.html#D10" target="_self" title="TOPGEN_1:def.10">Def10</a></i>;<br/>
<a NAME="E12:71"/>
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>  holds <br/>( <font color="Olive" title="b1">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">CC</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Olive" title="b1">B</font> is <a href="topgen_1.html#V2" title="TOPGEN_1:attr.2">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" title="c4">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><a NAME="E2:71_1"/><i><font color="Green" title="E45">A7</font></i>: 
( not <font color="Maroon" title="c4">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">CC</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c4">B</font> is <a href="topgen_1.html#V2" title="TOPGEN_1:attr.2">dense-in-itself</a> )
 ;<br/>

<a NAME="E3:71_1"/><i><font color="Green" title="E46">A8</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">CC</font>
 
<b>by </b><i><a class="txt" href="topgen_1.html#E2:71_1"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/>
<a NAME="E4:71_1"/>
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">CC</font>
 
<b>by </b><i><a class="txt" href="topgen_1.html#E3:71"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="topgen_1.html#E2:71_1"><i><font color="Green" title="E45">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:71_1"/>
contradiction
 <b>by </b><i><a class="txt" href="topgen_1.html#E2:71_1"><i><font color="Green" title="E45">A7</font></i></a>, <a class="txt" href="topgen_1.html#E3:71_1"><i><font color="Green" title="E46">A8</font></i></a>, <a class="ref" href="xboole_1.html#T69" title="XBOOLE_1:th.69">XBOOLE_1:69</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:71"/><b>then </b><i><font color="Green" title="E45">A9</font></i>: 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">CC</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="topgen_1.html#V6" title="TOPGEN_1:attr.6">scattered</a>
 
<b>by </b><i><a class="ref" href="topgen_1.html#D11" target="_self" title="TOPGEN_1:def.11">Def11</a></i>;<br/>
<a NAME="E14:71"/><i><font color="Green" title="E46">A10</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">CC</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">CC</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></i>;<br/>
<a NAME="E15:71"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">CC</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">CC</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<b>hence </b><a NAME="E16:71"/>
 ex <font color="Olive" title="b1">A</font>, <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 <br/>( <font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> &amp; <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b1">A</font> is <a href="topgen_1.html#V5" title="TOPGEN_1:attr.5">perfect</a> &amp; <font color="Olive" title="b2">B</font> is <a href="topgen_1.html#V6" title="TOPGEN_1:attr.6">scattered</a> )
 <b>by </b><i><a class="txt" href="topgen_1.html#E11:71"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="topgen_1.html#E13:71"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="topgen_1.html#E14:71"><i><font color="Green" title="E46">A10</font></i></a></i>;<br/>


</div>
