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<b>let </b><font color="Maroon" title="c1">PT</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">PT</font> is <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a> implies <font color="Maroon" title="c1">PT</font> is <a href="pcomps_1.html#V2" title="PCOMPS_1:attr.2">paracompact</a> )</span><br/>

<b>assume </b><a NAME="E1:13"/>
<font color="Maroon" title="c1">PT</font> is <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a>
 ; <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="E2:13"/><b>then </b>
for <font color="Olive" title="b1">FX</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 <font color="Olive" title="b1">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="Olive" title="b1">FX</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> holds <br/> ex <font color="Olive" title="b2">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="b2">GX</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> &amp; <font color="Olive" title="b2">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="b2">GX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b1">FX</font> &amp; <font color="Olive" title="b2">GX</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a> )
 
<b>by </b><i><a class="ref" href="pcomps_2.html#T12" target="_self" title="PCOMPS_2:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E3:13"/>
<font color="Maroon" title="c1">PT</font> is <a href="pcomps_1.html#V2" title="PCOMPS_1:attr.2">paracompact</a>
 <b>by </b><i><a class="ref" href="pcomps_1.html#D4" title="PCOMPS_1:def.4">PCOMPS_1:def 4</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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