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<b>let </b><font color="Maroon" title="c1">T</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ;<br/>

<b>thus </b><a NAME="E1:7"/>
( <font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> implies  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D7" target="_self" title="COMPTS_1:def.7">COMPTS_1:def 7</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:7_1"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 <b>and </b><br/><a NAME="E3:7_1"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 ;<br/>

<a NAME="E4:7_1"/>
(  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">F</font> )
 
<b>by </b><i><a class="txt" href="compts_1.html#E2:7_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self" title="COMPTS_1:def.1">Def1</a></i>;<br/>
<a NAME="E5:7_1"/><b>then </b>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E6:7_1"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D8" title="PRE_TOPC:def.8">PRE_TOPC:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">G</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E8:7_1"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c3">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
 <b>and </b><br/><a NAME="E9:7_1"/><i><font color="Green" title="E12">A5</font></i>: 
<font color="Maroon" title="c3">G</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">T</font>
 <b>and </b><br/><a NAME="E10:7_1"/><i><font color="Green" title="E13">A6</font></i>: 
<font color="Maroon" title="c3">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="compts_1.html#E1:7_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="compts_1.html#E3:7_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="compts_1.html#D3" target="_self" title="COMPTS_1:def.3">Def3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">G</font>
;<br/>

<a NAME="E11:7_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E9:7_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="pre_topc.html#D8" title="PRE_TOPC:def.8">PRE_TOPC:def 8</a></i>;<br/>
<b>hence </b><a NAME="E12:7_1"/>
( <font color="Maroon" title="c3">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c3">G</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c3">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="compts_1.html#E8:7_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="compts_1.html#E10:7_1"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self" title="COMPTS_1:def.1">Def1</a></i>;<br/>


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

<b>assume </b><a NAME="E2:7"/><i><font color="Green" title="E8">A7</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D3" target="_self" title="COMPTS_1:def.3">COMPTS_1:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E3:7"/><i><font color="Green" title="E9">A8</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">T</font>
 <b>and </b><br/><a NAME="E4:7"/><i><font color="Green" title="E10">A9</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 ;<br/>

<a NAME="E5:7"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E3:7"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="pre_topc.html#D8" title="PRE_TOPC:def.8">PRE_TOPC:def 8</a></i>;<br/>
<a NAME="E6:7"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="compts_1.html#D1" target="_self" title="COMPTS_1:def.1">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">G</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E8:7"/><i><font color="Green" title="E11">A10</font></i>: 
<font color="Maroon" title="c3">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
 <b>and </b><br/><a NAME="E9:7"/><i><font color="Green" title="E12">A11</font></i>: 
<font color="Maroon" title="c3">G</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 <b>and </b><br/><a NAME="E10:7"/><i><font color="Green" title="E13">A12</font></i>: 
<font color="Maroon" title="c3">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="compts_1.html#E2:7"><i><font color="Green" title="E8">A7</font></i></a>, <a class="txt" href="compts_1.html#E4:7"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="compts_1.html#D7" target="_self" title="COMPTS_1:def.7">Def7</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">G</font>
;<br/>

<a NAME="E11:7"/>
(  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">G</font> )
 
<b>by </b><i><a class="txt" href="compts_1.html#E9:7"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self" title="COMPTS_1:def.1">Def1</a></i>;<br/>
<a NAME="E12:7"/><b>then </b>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E13:7"/>
( <font color="Maroon" title="c3">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c3">G</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c3">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="compts_1.html#E8:7"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="compts_1.html#E10:7"><i><font color="Green" title="E13">A12</font></i></a>, <a class="ref" href="pre_topc.html#D8" title="PRE_TOPC:def.8">PRE_TOPC:def 8</a></i>;<br/>


</div>
