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<b>let </b><font color="Maroon">c<sub>1</sub></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="E1:13"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="pcomps_1.html#V3">metrizable</a>
 ;<br/>

<a NAME="E2:13"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tops_2.html#V1">open</a> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> is <a href="tops_2.html#V1">open</a> &amp; <font color="Olive">b<sub>2</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pcomps_1.html#V1">locally_finite</a> )
 
<b>by </b><i><a class="ref" href="pcomps_2.html#T12" target="_self">Th12</a></i>;<br/>
<b>hence </b><a NAME="E3:13"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="pcomps_1.html#V2">paracompact</a>
 <b>by </b><i><a class="ref" href="pcomps_1.html#D4">PCOMPS_1:def 4</a></i>;<br/>


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