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<span class="kw">let </span><font color="Maroon" title="c2">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">PT</font> is  <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a>  implies <font color="Maroon" title="c2">PT</font> is  <a href="pcomps_1.html#V2" title="PCOMPS_1:attr.2">paracompact</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:12"/>
<font color="Maroon" title="c2">PT</font> is  <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">PT</font> is  <a href="pcomps_1.html#V2" title="PCOMPS_1:attr.2">paracompact</a> </span><br/>

<a NAME="E2:12"/><span class="kw">then </span>
 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="c2">PT</font>  st <font color="Olive" title="b1">FX</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c2">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="c2">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#NM13" title="STRUCT_0:NM.13">Cover</a> of <font color="Maroon" title="c2">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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcomps_2.html#T6" target="_self" onmouseover="rs('pcomps_2/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:12"/>
<font color="Maroon" title="c2">PT</font> is  <a href="pcomps_1.html#V2" title="PCOMPS_1:attr.2">paracompact</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pcomps_1.html#D3" onmouseover="rs('pcomps_1/D3')" onmouseout="rh()">PCOMPS_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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