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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>let </b><font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c3">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:16"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c2">P</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c3">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">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="E3:16"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</font>
 <b>and </b><br/><a NAME="E4:16"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c4">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 ;<br/>

<a NAME="E5:16"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</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="c4">F</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E3:16"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self" title="COMPTS_1:def.1">Def1</a></i>;<br/>
<a NAME="E6:16"/>
<font color="Maroon" title="c2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E7:16"/><b>then </b>
<font color="Maroon" title="c2">P</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="c4">F</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E5:16"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:16"/><b>then </b>
<font color="Maroon" title="c4">F</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c2">P</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="c5">G1</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="E10:16"/><i><font color="Green" title="E20">A6</font></i>: 
<font color="Maroon" title="c5">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">F</font>
 <b>and </b><br/><a NAME="E11:16"/><i><font color="Green" title="E21">A7</font></i>: 
<font color="Maroon" title="c5">G1</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c2">P</font>
 <b>and </b><br/><a NAME="E12:16"/><i><font color="Green" title="E22">A8</font></i>: 
<font color="Maroon" title="c5">G1</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#E1:16"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="compts_1.html#E4:16"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="compts_1.html#D7" target="_self" title="COMPTS_1:def.7">Def7</a></i>;<br/>
<a NAME="E13:16"/>
<font color="Maroon" title="c3">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E14:16"/><b>then </b>
<font color="Maroon" title="c3">Q</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="c4">F</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E5:16"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E15:16"/><b>then </b>
<font color="Maroon" title="c4">F</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c3">Q</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="c6">G2</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="E17:16"/><i><font color="Green" title="E23">A9</font></i>: 
<font color="Maroon" title="c6">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">F</font>
 <b>and </b><br/><a NAME="E18:16"/><i><font color="Green" title="E24">A10</font></i>: 
<font color="Maroon" title="c6">G2</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c3">Q</font>
 <b>and </b><br/><a NAME="E19:16"/><i><font color="Green" title="E25">A11</font></i>: 
<font color="Maroon" title="c6">G2</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#E2:16"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="compts_1.html#E4:16"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="compts_1.html#D7" target="_self" title="COMPTS_1:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">G</font> = <font color="Maroon" title="c5">G1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">G2</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c7">G</font>
;<br/>

<b>thus </b><a NAME="E21:16"/>
<font color="Maroon" title="c7">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E10:16"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="compts_1.html#E17:16"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>

<a NAME="E22:16"/>
( <font color="Maroon" title="c2">P</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="c5">G1</font> &amp; <font color="Maroon" title="c3">Q</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="c6">G2</font> )
 
<b>by </b><i><a class="txt" href="compts_1.html#E11:16"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="compts_1.html#E18:16"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self" title="COMPTS_1:def.1">Def1</a></i>;<br/>
<a NAME="E23:16"/><b>then </b>
<font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">G1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c6">G2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T13" title="XBOOLE_1:th.13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E24:16"/><b>then </b>
<font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">G2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></i>;<br/>
<b>hence </b><a NAME="E25:16"/>
<font color="Maroon" title="c7">G</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</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>thus </b><a NAME="E26:16"/>
<font color="Maroon" title="c7">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#E12:16"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="compts_1.html#E19:16"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="finset_1.html#T14" title="FINSET_1:th.14">FINSET_1:14</a></i>;<br/>


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