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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/>

<b>thus </b><a NAME="E1:7"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="compts_1.html#V2" target="_self">compact</a> implies  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="compts_1.html#V6" target="_self">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">E8</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="compts_1.html#V2" target="_self">compact</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D7" target="_self">COMPTS_1:def 7</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:7_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="compts_1.html#R1" target="_self">is_a_cover_of</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:7_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_2.html#V1">open</a>
 ;<br/>

<a NAME="E4:7_1"/>
(  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="compts_1.html#E2:7_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E5:7_1"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E6:7_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:7_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E9:7_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E10:7_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="compts_1.html#E1:7_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="compts_1.html#E3:7_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="compts_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<a NAME="E11:7_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E9:7_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/>
<b>hence </b><a NAME="E12:7_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="compts_1.html#R1" target="_self">is_a_cover_of</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="txt" href="compts_1.html#E8:7_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="compts_1.html#E10:7_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self">Def1</a></i>;<br/>


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

<b>assume </b><a NAME="E2:7"/><i><font color="Green">E8</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="compts_1.html#V6" target="_self">compact</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D3" target="_self">COMPTS_1:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E3:7"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:7"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_2.html#V1">open</a>
 ;<br/>

<a NAME="E5:7"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E3:7"><i><font color="Green">E9</font></i></a>, <a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/>
<a NAME="E6:7"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="compts_1.html#R1" target="_self">is_a_cover_of</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="compts_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:7"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E9:7"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="compts_1.html#R1" target="_self">is_a_cover_of</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E10:7"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="compts_1.html#E2:7"><i><font color="Green">E8</font></i></a>, <a class="txt" href="compts_1.html#E4:7"><i><font color="Green">E10</font></i></a>, <a class="ref" href="compts_1.html#D7" target="_self">Def7</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<a NAME="E11:7"/>
(  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="compts_1.html#E9:7"><i><font color="Green">E12</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E12:7"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E13:7"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="txt" href="compts_1.html#E8:7"><i><font color="Green">E11</font></i></a>, <a class="txt" href="compts_1.html#E10:7"><i><font color="Green">E13</font></i></a>, <a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/>


</div>
