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<b>let </b><font color="Maroon">T</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:14"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">T</font> is <a href="compts_1.html#V2" target="_self">compact</a>
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</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="E3:14"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F</font> <a href="compts_1.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E4:14"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">pp</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<b>set </b><font color="Maroon">FP</font> = <font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">pp</font>;<br/>
<a NAME="E6:14"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:14"/><b>then </b>
<font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/>
<a NAME="E8:14"/><b>then </b>
(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T18">PRE_TOPC:18</a></i>;<br/>
<b>then </b> <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>
<br/>.= 
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T96">ZFMISC_1:96</a></i>
;<br/>
<a NAME="E10:14"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">pp</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/>
<a NAME="E11:14"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<a NAME="E12:14"/>
<font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">pp</font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">H</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:14_2"/>
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">pp</font>
 ;<br/>

<a NAME="E2:14_2"/><b>then </b>
( <font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> or <font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:14_2"/>
<font color="Maroon">H</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="compts_1.html#D1" target="_self">Def1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E14:14"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">pp</font>
 <b>and </b><br/><a NAME="E15:14"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">G</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E16:14"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">G'</font> = <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">pp</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<b>take </b>
<font color="Maroon">G'</font>
;<br/>

<a NAME="E18:14"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">G'</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="compts_1.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/>
<a NAME="E19:14"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>;<br/>
<a NAME="E20:14"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<b>hence </b><a NAME="E21:14"/>
<font color="Maroon">G'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="compts_1.html#D4" target="_self">Def4</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<b>reconsider </b><font color="Maroon">M</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">M</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>
<br/>.= 
<font color="Maroon">P</font>

;<br/>
<a NAME="E24:14"/><b>then </b>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G'</font>
 
<b>by </b><i><a class="ref" href="tops_2.html#T6">TOPS_2:6</a></i>;<br/>
<b>hence </b><a NAME="E25:14"/>
<font color="Maroon">G'</font> <a href="compts_1.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E26:14"/>
<font color="Maroon">G'</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="compts_1.html#D3" target="_self">Def3</a>, <a class="ref" href="finset_1.html#T16">FINSET_1:16</a></i>;<br/>


</div>
