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<b>let </b><font color="Maroon">T</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="E1:26"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">T</font> is <a href="compts_1.html#V2">compact</a>
 ;<br/>

<a NAME="E2:26"/>
for <font color="Olive">FX</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>  st <font color="Olive">FX</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">T</font> &amp; <font color="Olive">FX</font> is <a href="tops_2.html#V1">open</a> holds <br/> ex <font color="Olive">GX</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">GX</font> is <a href="tops_2.html#V1">open</a> &amp; <font color="Olive">GX</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">T</font> &amp; <font color="Olive">GX</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">FX</font> &amp; <font color="Olive">GX</font> is <a href="pcomps_1.html#V1" target="_self">locally_finite</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">FX</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:26_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">FX</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">T</font> &amp; <font color="Maroon">FX</font> is <a href="tops_2.html#V1">open</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">GX</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:26_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">GX</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FX</font> &amp; <font color="Maroon">GX</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">T</font> &amp; <font color="Maroon">GX</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i>, <a class="ref" href="compts_1.html#D3">COMPTS_1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">GX</font>
;<br/>

<b>thus </b><a NAME="E4:26_1"/>
<font color="Maroon">GX</font> is <a href="tops_2.html#V1">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1"/>
for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> holds <br/><font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, , <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:26_1_1"/>
<font color="Maroon">GX</font> is <a href="tops_2.html#V1">open</a>
 <b>by </b><i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E5:26_1"/>
<font color="Maroon">GX</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">T</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E6:26_1"/>
<font color="Maroon">GX</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">FX</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#T17">SETFAM_1:17</a></i>;<br/>

<b>thus </b><a NAME="E7:26_1"/>
<font color="Maroon">GX</font> is <a href="pcomps_1.html#V1" target="_self">locally_finite</a>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:26"/>
<font color="Maroon">T</font> is <a href="pcomps_1.html#V2" target="_self">paracompact</a>
 <b>by </b><i/>;<br/>


</div>
