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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="compts_1.html#V2">compact</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="compts_1.html#V2">compact</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>





<b>assume </b><a NAME="E1:35"/>
( <font color="Maroon">F</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> &amp; <font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a> &amp; <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">Q</font> where <font color="Olive">Q</font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> :  ex <font color="Olive">FQ</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> st <br/>( <font color="Olive">FQ</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp; <font color="Olive">FQ</font> is <a href="finset_1.html#V1">finite</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>,<font color="Olive">Q</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Olive">FQ</font> ) </span>}</span>  )
 ;<br/>

<a NAME="E2:35"/><b>then </b>
( <font color="Maroon">R</font> is <a href="tops_2.html#V1">open</a> &amp; <font color="Maroon">R</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">X</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:35"/><b>then </b>
 ex <font color="Olive">G</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> &amp; <font color="Olive">G</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">X</font> &amp; <font color="Olive">G</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>hence </b><a NAME="E4:35"/>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> &amp; <font color="Olive">C</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">X</font> )
 ;<br/>


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