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<b>let </b><font color="Maroon" title="c1">T</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>;<br/>

<b>thus </b><a NAME="E1:46"/>
( <font color="Maroon" title="c1">T</font> is <a href="compl_sp.html#V8" title="COMPL_SP:attr.8">countably_compact</a> implies for <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">F</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a> &amp; ( for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) holds <br/><font color="Olive" title="b1">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1"/>
<font color="Maroon" title="c1">T</font> is <a href="compl_sp.html#V8" title="COMPL_SP:attr.8">countably_compact</a>
 ;<br/>

<a NAME="E2:46_1"/><b>then </b>
for <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">F</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a> &amp; <font color="Olive" title="b1">F</font> is <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a> holds <br/><font color="Olive" title="b1">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="compl_sp.html#T26" target="_self" title="COMPL_SP:th.26">Th26</a></i>;<br/>
<b>hence </b><a NAME="E3:46_1"/>
for <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">F</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a> &amp; ( for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) holds <br/><font color="Olive" title="b1">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="compl_sp.html#E35"><i><font color="Green" title="E32">Lm3</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:46"/>
( ( for <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">F</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a> &amp; ( for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) holds <br/><font color="Olive" title="b1">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) implies <font color="Maroon" title="c1">T</font> is <a href="compl_sp.html#V8" title="COMPL_SP:attr.8">countably_compact</a> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E40"><i><font color="Green" title="E37">Lm6</font></i></a></i>;<br/>


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