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<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ;<br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c4">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:40"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E2:40"/><i><font color="Green" title="E12">A2</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T</font> #)
 <b>and </b><br/><a NAME="E3:40"/><i><font color="Green" title="E13">A3</font></i>: 
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">S</font>  st <font color="Olive" title="b1">F</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b1">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> holds <br/> ex <font color="Olive" title="b2">G</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b2">G</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D7" title="COMPTS_1:def.7">COMPTS_1:def 7</a><br/>

<b>let </b><font color="Maroon" title="c5">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D7" title="COMPTS_1:def.7">COMPTS_1:def 7</a><br/>

<b>assume </b><a NAME="E4:40"/><i><font color="Green" title="E14">A4</font></i>: 
( <font color="Maroon" title="c5">F</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c5">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c6">K</font> = <font color="Maroon" title="c5">F</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="yellow14.html#E2:40"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/>
<a NAME="E6:40"/><i><font color="Green" title="E15">A5</font></i>: 
<font color="Maroon" title="c6">K</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c3">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:40_1"/>
<font color="Maroon" title="c3">A</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">K</font>
 <b>by </b><i><a class="txt" href="yellow14.html#E1:40"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="yellow14.html#E4:40"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="compts_1.html#D1" title="COMPTS_1:def.1">COMPTS_1:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D1" title="COMPTS_1:def.1">COMPTS_1:def 1</a><br/>


</div><b>end;</b></div>
<a NAME="E7:40"/>
<font color="Maroon" title="c6">K</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 
<b>by </b><i><a class="txt" href="yellow14.html#E2:40"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="yellow14.html#E4:40"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="waybel_9.html#T19" title="WAYBEL_9:th.19">WAYBEL_9:19</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">L</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E9:40"/><i><font color="Green" title="E16">A6</font></i>: 
( <font color="Maroon" title="c7">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">K</font> &amp; <font color="Maroon" title="c7">L</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c7">L</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="yellow14.html#E3:40"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="yellow14.html#E6:40"><i><font color="Green" title="E15">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">G</font> = <font color="Maroon" title="c7">L</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">T</font> <b>by </b><i><a class="txt" href="yellow14.html#E2:40"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">G</font>
;<br/>

<a NAME="E11:40"/>
<font color="Maroon" title="c8">G</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c4">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:40_2"/>
<font color="Maroon" title="c4">B</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="c8">G</font>
 <b>by </b><i><a class="txt" href="yellow14.html#E1:40"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="yellow14.html#E9:40"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="compts_1.html#D1" title="COMPTS_1:def.1">COMPTS_1:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D1" title="COMPTS_1:def.1">COMPTS_1:def 1</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:40"/>
( <font color="Maroon" title="c8">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Maroon" title="c8">G</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c8">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="yellow14.html#E9:40"><i><font color="Green" title="E16">A6</font></i></a></i>;<br/>


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