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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:20"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="compts_1.html#V2" target="_self">compact</a>
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:20"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E3:20"/><i><font color="Green">E17</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D3" target="_self">COMPTS_1:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E4:20"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E5:20"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="tops_2.html#V1">open</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="funct_3.html#K3">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tops_2.html#T54">TOPS_2:54</a></i>;<br/>
<a NAME="E7:20"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="compts_1.html#E3:20"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>then </b><i><font color="Green">E20</font></i>:  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_3.html#T30">FUNCT_3:30</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="compts_1.html#E3:20"><i><font color="Green">E17</font></i></a>, <a class="txt" href="compts_1.html#E4:20"><i><font color="Green">E18</font></i></a>, <a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E9:20"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:20"/><b>then </b>
(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T146">FUNCT_1:146</a></i>;<br/>
<a NAME="E11:20"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E8:20"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E12:20"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/>
<a NAME="E13:20"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="tops_2.html#V1">open</a>
 
<b>by </b><i><a class="txt" href="compts_1.html#E2:20"><i><font color="Green">E16</font></i></a>, <a class="txt" href="compts_1.html#E5:20"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tops_2.html#T59">TOPS_2:59</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</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="E15:20"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E16:20"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>7</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="E17:20"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="compts_1.html#E1:20"><i><font color="Green">E15</font></i></a>, <a class="txt" href="compts_1.html#E12:20"><i><font color="Green">E21</font></i></a>, <a class="ref" href="compts_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="funct_3.html#K2">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<a NAME="E19:20"/><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K2">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K2">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E15:20"><i><font color="Green">E22</font></i></a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E20:20"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K3">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K2">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="funct_3.html#T33">FUNCT_3:33</a></i>;<br/>
<a NAME="E21:20"/><b>then </b>
( <span class="p1">(<span class="default"><a href="funct_3.html#K2">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K3">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K2">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K2">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="funct_3.html#K2">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K2">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T145">FUNCT_1:145</a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E22:20"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_3.html#K2">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E23:20"/>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="compts_1.html#E19:20"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<a NAME="E24:20"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E25:20"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
;<br/>
<b>then </b> <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_3.html#T16">FUNCT_3:16</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="compts_1.html#E16:20"><i><font color="Green">E23</font></i></a>, <a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="compts_1.html#E3:20"><i><font color="Green">E17</font></i></a>, <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>
;<br/>
<b>hence </b><a NAME="E27:20"/>
<font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/>

<b>thus </b><a NAME="E28:20"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="compts_1.html#E17:20"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finset_1.html#T17">FINSET_1:17</a></i>;<br/>


</div>
