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<b>let </b><font color="Maroon" title="c1">T</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c3">S</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">S</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:21"/><i><font color="Green" title="E16">A1</font></i>: 
( <font color="Maroon" title="c4">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">S</font> )
 <b>and </b><br/><a NAME="E2:21"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c2">P</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</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="c3">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D7" target="_self" title="COMPTS_1:def.7">COMPTS_1:def 7</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E3:21"/><i><font color="Green" title="E18">A3</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">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font>
 <b>and </b><br/><a NAME="E4:21"/><i><font color="Green" title="E19">A4</font></i>: 
<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">G</font> = <span class="p1">(<span class="default"><a href="funct_3.html#K3" title="FUNCT_3:func.3">"</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <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">T</font> <b>by </b><i><a class="ref" href="tops_2.html#T54" title="TOPS_2:th.54">TOPS_2:54</a></i>;<br/>
<a NAME="E6:21"/>
<font color="Maroon" title="c5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="compts_1.html#E1:21"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/>
<a NAME="E7:21"/><b>then </b><i><font color="Green" title="E20">A5</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_3.html#T30" title="FUNCT_3:th.30">FUNCT_3:30</a></i>;<br/>
<a NAME="E8:21"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</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="c5">F</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E3:21"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self" title="COMPTS_1:def.1">Def1</a></i>;<br/>
<a NAME="E9:21"/><b>then </b><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T178" title="RELAT_1:th.178">RELAT_1:178</a></i>;<br/>
<a NAME="E10:21"/>
<font color="Maroon" title="c2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 
;<br/>
<a NAME="E11:21"/><b>then </b>
<font color="Maroon" title="c2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:21"/><b>then </b>
<font color="Maroon" title="c2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T146" title="FUNCT_1:th.146">FUNCT_1:146</a></i>;<br/>
<a NAME="E13:21"/><b>then </b>
<font color="Maroon" title="c2">P</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">G</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E7:21"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="compts_1.html#E9:21"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E14:21"/><b>then </b><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c6">G</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c2">P</font>
 
<b>by </b><i><a class="ref" href="compts_1.html#D1" target="_self" title="COMPTS_1:def.1">Def1</a></i>;<br/>
<a NAME="E15:21"/>
<font color="Maroon" title="c6">G</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 
<b>by </b><i><a class="txt" href="compts_1.html#E1:21"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="compts_1.html#E4:21"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="tops_2.html#T59" title="TOPS_2:th.59">TOPS_2:59</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">G'</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E17:21"/><i><font color="Green" title="E23">A8</font></i>: 
<font color="Maroon" title="c7">G'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">G</font>
 <b>and </b><br/><a NAME="E18:21"/><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon" title="c7">G'</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c2">P</font>
 <b>and </b><br/><a NAME="E19:21"/><i><font color="Green" title="E25">A10</font></i>: 
<font color="Maroon" title="c7">G'</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="compts_1.html#E2:21"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="compts_1.html#E14:21"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="compts_1.html#D7" target="_self" title="COMPTS_1:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">F'</font> = <span class="p1">(<span class="default"><a href="funct_3.html#K2" title="FUNCT_3:func.2">.:</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">G'</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c3">S</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c8">F'</font>
;<br/>

<a NAME="E21:21"/><i><font color="Green" title="E26">A11</font></i>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K2" title="FUNCT_3:func.2">.:</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">G'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K2" title="FUNCT_3:func.2">.:</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">G</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E17:21"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<a NAME="E22:21"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K3" title="FUNCT_3:func.3">"</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K2" title="FUNCT_3:func.2">.:</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c5">F</font>
 
<b>by </b><i><a class="ref" href="funct_3.html#T33" title="FUNCT_3:th.33">FUNCT_3:33</a></i>;<br/>
<a NAME="E23:21"/><b>then </b>
( <span class="p1">(<span class="default"><a href="funct_3.html#K2" title="FUNCT_3:func.2">.:</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K3" title="FUNCT_3:func.3">"</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K2" title="FUNCT_3:func.2">.:</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K2" title="FUNCT_3:func.2">.:</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c5">F</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="funct_3.html#K2" title="FUNCT_3:func.2">.:</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K2" title="FUNCT_3:func.2">.:</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">F</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T145" title="FUNCT_1:th.145">FUNCT_1:145</a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<a NAME="E24:21"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_3.html#K2" title="FUNCT_3:func.2">.:</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">F</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E25:21"/>
<font color="Maroon" title="c8">F'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E21:21"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>

<a NAME="E26:21"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E27:21"/><b>then </b>
<font color="Maroon" title="c7">G'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
;<br/>
<a NAME="E28:21"/><b>then </b><i><font color="Green" title="E27">A12</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">F'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">G'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_3.html#T16" title="FUNCT_3:th.16">FUNCT_3:16</a></i>;<br/>
<a NAME="E29:21"/>
<font color="Maroon" title="c2">P</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="c7">G'</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E18:21"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self" title="COMPTS_1:def.1">Def1</a></i>;<br/>
<a NAME="E30:21"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</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">F'</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E28:21"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<b>hence </b><a NAME="E31:21"/>
<font color="Maroon" title="c8">F'</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="ref" href="compts_1.html#D1" target="_self" title="COMPTS_1:def.1">Def1</a></i>;<br/>

<b>thus </b><a NAME="E32:21"/>
<font color="Maroon" title="c8">F'</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="compts_1.html#E19:21"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="finset_1.html#T17" title="FINSET_1:th.17">FINSET_1:17</a></i>;<br/>


</div>
