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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="pre_topc.html#NM1">TopSpace</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>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:21"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_2.html#V2" target="_self">closed</a>
 <b>and </b><br/><a NAME="E2:21"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E4:21"/>
(  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">c<sub>2</sub></font> is <a href="tops_2.html#V1" target="_self">open</a> &amp;  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="txt" href="tops_2.html#E1:21"><i><font color="Green">E10</font></i></a>, <a class="txt" href="tops_2.html#E2:21"><i><font color="Green">E11</font></i></a>, <a class="ref" href="tops_2.html#T13" target="_self">Th13</a>, <a class="ref" href="tops_2.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E5:21"/><b>then </b><i><font color="Green">E12</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T27" target="_self">Th27</a></i>;<br/>
<div><i><font color="Green">E13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:21_1"/><b>then </b>
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="tops_2.html#T11" target="_self">Th11</a></i>;<br/><b>hence </b><a NAME="E3:21_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="tops_2.html#E5:21"><i><font color="Green">E12</font></i></a>, <a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:21_2"/><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#K1">{}</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/><b>hence </b><a NAME="E3:21_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T22">TOPS_1:22</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:21"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="tops_2.html#E6:21"><i><font color="Green">E13</font></i></a></i>;<br/>


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