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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V7" title="PRE_TOPC:attr.7">T_1</a>  implies  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>  ) )</span><br/>

<span class="kw">defpred </span><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>] <span class="kw">means </span>$1 is  <a href="topgen_1.html#V2" title="TOPGEN_1:attr.2">dense-in-itself</a> ;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E2:71"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
 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>] )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="subset_1.html#S3" title="SUBSET_1:sch.3">SUBSET_1:sch 3</a>();<br/></span>
<span class="kw">set </span><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="E3:71"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
(  <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> &amp;  <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T2" title="PRE_TOPC:th.2">PRE_TOPC:2</a>, <a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/>
<a NAME="E4:71"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<font color="Maroon" title="c2">CC</font> is  <a href="topgen_1.html#V4" title="TOPGEN_1:attr.4">dense-in-itself</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="topgen_1.html#D9" target="_self" title="TOPGEN_1:def.9">Def9</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:71"/>
<font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V7" title="PRE_TOPC:attr.7">T_1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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>  )</span><br/>

<a NAME="E6:71"/><span class="kw">then </span>
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:71"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="topgen_1.html#T36" target="_self" title="TOPGEN_1:th.36">Th38</a>, <a class="ref" href="topgen_1.html#T38" target="_self" title="TOPGEN_1:th.38">Th40</a></span>;<br/>
<a NAME="E7:71"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E39">A1</font></span></a></span>;<br/>
<a NAME="E8:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A4</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></span>;<br/>
<a NAME="E10:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A5</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:71"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E11:71"/>
 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"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><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><span class="kw"> such that </span><a NAME="E2:71_1"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
 not <font color="Maroon" title="c4">B</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">and </span><br/><a NAME="E3:71_1"/><span class="lab"><font color="Green" title="E45">A7</font></span>: 
( <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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:71_1"/>
( <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> &amp; <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E3:71_1"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="subset_1.html#T23" title="SUBSET_1:th.23">SUBSET_1:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:71_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1"><span class="lab"><font color="Green" title="E44">A6</font></span></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#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A8</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_1.html#D11" target="_self" title="TOPGEN_1:def.11">Def11</a></span>;<br/>
<a NAME="E13:71"/>
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:71"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="topgen_1.html#T38" target="_self" title="TOPGEN_1:th.38">Th40</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:71"/>
 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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:71"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E12:71"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="txt" href="#E3:71"><span class="lab"><font color="Green" title="E40">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
