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<b>take </b><font color="Maroon">c<sub>1</sub></font> =  <a href="compts_1.html#K1">1TopSp</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>;<br/>

<a NAME="E1:25_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T9" target="_self">Th9</a></i>;<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="pcomps_1.html#D4" target="_self">PCOMPS_1:def 4</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:25_1_1"/><i><font color="Green">E21</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="E3:25_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_2.html#V1">open</a>
 ;<br/>

<b>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="E5:25_1_1"/><i><font color="Green">E23</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="E6:25_1_1"/><i><font color="Green">E24</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="E7:25_1_1"/><i><font color="Green">E25</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="pcomps_1.html#E1:25_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="pcomps_1.html#E2:25_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pcomps_1.html#E3:25_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="compts_1.html#D3">COMPTS_1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E8:25_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="tops_2.html#V1">open</a> &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> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="pcomps_1.html#V1" target="_self">locally_finite</a> )
 <b>by </b><i><a class="txt" href="pcomps_1.html#E3:25_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="pcomps_1.html#E5:25_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="pcomps_1.html#E6:25_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pcomps_1.html#E7:25_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="pcomps_1.html#T13" target="_self">Th13</a>, <a class="ref" href="setfam_1.html#T17">SETFAM_1:17</a>, <a class="ref" href="tops_2.html#T18">TOPS_2:18</a></i>;<br/>


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