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<b>take </b><font color="Maroon" title="c1">PT</font> =  <a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">PT</font> is <a href="pcomps_1.html#V2" title="PCOMPS_1:attr.2">paracompact</a></span><br/>

<a NAME="E1:24_1_1"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c1">PT</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T9" target="_self" title="PCOMPS_1:th.9">Th9</a></i>;<br/>
<b>let </b><font color="Maroon" title="c2">FX</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">PT</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pcomps_1.html#D4" target="_self" title="PCOMPS_1:def.4">PCOMPS_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">FX</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <font color="Maroon" title="c1">PT</font> &amp; <font color="Maroon" title="c2">FX</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> implies  ex <font color="Olive" title="b1">GX</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b1">GX</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> &amp; <font color="Olive" title="b1">GX</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <font color="Maroon" title="c1">PT</font> &amp; <font color="Olive" title="b1">GX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c2">FX</font> &amp; <font color="Olive" title="b1">GX</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a> ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:24_1_1"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c2">FX</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <font color="Maroon" title="c1">PT</font>
 <b>and </b><br/><a NAME="E3:24_1_1"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c2">FX</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">GX</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">PT</font> st <br/>( <font color="Olive" title="b1">GX</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> &amp; <font color="Olive" title="b1">GX</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <font color="Maroon" title="c1">PT</font> &amp; <font color="Olive" title="b1">GX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c2">FX</font> &amp; <font color="Olive" title="b1">GX</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a> )</span><br/>

<b>consider </b><font color="Maroon" title="c3">GX</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">PT</font><b> such that </b><br/><a NAME="E5:24_1_1"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c3">GX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">FX</font>
 <b>and </b><br/><a NAME="E6:24_1_1"/><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c3">GX</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <font color="Maroon" title="c1">PT</font>
 <b>and </b><br/><a NAME="E7:24_1_1"/><i><font color="Green" title="E24">A6</font></i>: 
<font color="Maroon" title="c3">GX</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E1:24_1_1"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="pcomps_1.html#E2:24_1_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="pcomps_1.html#E3:24_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="compts_1.html#D3" title="COMPTS_1:def.3">COMPTS_1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">GX</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">GX</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> &amp; <font color="Maroon" title="c3">GX</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <font color="Maroon" title="c1">PT</font> &amp; <font color="Maroon" title="c3">GX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c2">FX</font> &amp; <font color="Maroon" title="c3">GX</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a> )</span><br/>

<b>thus </b><a NAME="E8:24_1_1"/>
( <font color="Maroon" title="c3">GX</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> &amp; <font color="Maroon" title="c3">GX</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <font color="Maroon" title="c1">PT</font> &amp; <font color="Maroon" title="c3">GX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c2">FX</font> &amp; <font color="Maroon" title="c3">GX</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a> )
 <b>by </b><i><a class="txt" href="pcomps_1.html#E3:24_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="pcomps_1.html#E5:24_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="pcomps_1.html#E6:24_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="pcomps_1.html#E7:24_1_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="pcomps_1.html#T13" target="_self" title="PCOMPS_1:th.13">Th13</a>, <a class="ref" href="setfam_1.html#T17" title="SETFAM_1:th.17">SETFAM_1:17</a>, <a class="ref" href="tops_2.html#T18" title="TOPS_2:th.18">TOPS_2:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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