<?xml version="1.0"?>
<div class="add">

<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">A</font> be    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c3">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>





<a NAME="E1:8"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">A</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>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D9" title="PRE_TOPC:def.9">PRE_TOPC:def 9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">AA</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>assume </b><a NAME="E3:8"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c3">Q</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="c2">A</font>
 ;<br/>

<a NAME="E4:8"/><b>then </b><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c3">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">AA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Q</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<b>thus </b><a NAME="E5:8"/>
( <font color="Maroon" title="c3">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> implies for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Q</font> holds <br/><font color="Olive" title="b1">P</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c3">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">A</font>;<br/>

<b>assume </b><a NAME="E2:8_1"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Q</font>
 ;<br/>

<b>thus </b><a NAME="E3:8_1"/>
<font color="Maroon" title="c5">P</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">G</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">A</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>set </b><font color="Maroon" title="c7">GG</font> = <font color="Maroon" title="c6">G</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:8_1_1"/><i><font color="Green" title="E13">A5</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="c5">P</font>
 <b>and </b><br/><a NAME="E2:8_1_1"/><i><font color="Green" title="E14">A6</font></i>: 
<font color="Maroon" title="c6">G</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">F</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="E4:8_1_1"/><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c8">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 <b>and </b><br/><a NAME="E5:8_1_1"/><i><font color="Green" title="E16">A8</font></i>: 
for <font color="Olive" title="b1">AA</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">AA</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="c2">A</font> holds <br/><font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F</font> <a href="tops_2.html#K1" title="TOPS_2:func.1">|</a> <font color="Olive" title="b1">AA</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E2:8_1_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="tops_2.html#T49" title="TOPS_2:th.49">TOPS_2:49</a></i>;<br/>
<a NAME="E6:8_1_1"/>
( <font color="Maroon" title="c5">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> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="tops_2.html#K1" title="TOPS_2:func.1">|</a> <font color="Maroon" title="c4">AA</font></span>)</span> <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#E1:8_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self" title="COMPTS_1:def.1">Def1</a>, <a class="ref" href="tops_2.html#T44" title="TOPS_2:th.44">TOPS_2:44</a></i>;<br/>
<a NAME="E7:8_1_1"/><b>then </b>
( <font color="Maroon" title="c3">Q</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> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c6">G</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#E2:8_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="compts_1.html#E5:8_1_1"><i><font color="Green" title="E16">A8</font></i></a></i>;<br/>
<a NAME="E8:8_1_1"/><b>then </b>
<font color="Maroon" title="c3">Q</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="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:8_1_1"/><b>then </b>
<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="c3">Q</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>then consider </b><font color="Maroon" title="c9">F'</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="E11:8_1_1"/><i><font color="Green" title="E17">A9</font></i>: 
<font color="Maroon" title="c9">F'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font>
 <b>and </b><br/><a NAME="E12:8_1_1"/><i><font color="Green" title="E18">A10</font></i>: 
<font color="Maroon" title="c9">F'</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c3">Q</font>
 <b>and </b><br/><a NAME="E13:8_1_1"/><i><font color="Green" title="E19">A11</font></i>: 
<font color="Maroon" title="c9">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#E1:8_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="compts_1.html#E4:8_1_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="compts_1.html#D7" target="_self" title="COMPTS_1:def.7">Def7</a></i>;<br/>
<a NAME="E14:8_1_1"/>
<font color="Maroon" title="c9">F'</font> <a href="tops_2.html#K1" title="TOPS_2:func.1">|</a> <font color="Maroon" title="c4">AA</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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">AA</font></span>)</span></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">G'</font> = <font color="Maroon" title="c9">F'</font> <a href="tops_2.html#K1" title="TOPS_2:func.1">|</a> <font color="Maroon" title="c4">AA</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">G'</font>
;<br/>

<a NAME="E16:8_1_1"/>
<font color="Maroon" title="c9">F'</font> <a href="tops_2.html#K1" title="TOPS_2:func.1">|</a> <font color="Maroon" title="c4">AA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="tops_2.html#K1" title="TOPS_2:func.1">|</a> <font color="Maroon" title="c4">AA</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E11:8_1_1"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="tops_2.html#T40" title="TOPS_2:th.40">TOPS_2:40</a></i>;<br/>
<b>hence </b><a NAME="E17:8_1_1"/>
<font color="Maroon" title="c10">G'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">G</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E5:8_1_1"><i><font color="Green" title="E16">A8</font></i></a></i>;<br/>

<a NAME="E18:8_1_1"/>
<font color="Maroon" title="c3">Q</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="c9">F'</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E12:8_1_1"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self" title="COMPTS_1:def.1">Def1</a></i>;<br/>
<a NAME="E19:8_1_1"/><b>then </b>
<font color="Maroon" title="c5">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="c10">G'</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E4:8"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="compts_1.html#E2:8_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="tops_2.html#T42" title="TOPS_2:th.42">TOPS_2:42</a></i>;<br/>
<b>hence </b><a NAME="E20:8_1_1"/>
<font color="Maroon" title="c10">G'</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c5">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="E21:8_1_1"/>
<font color="Maroon" title="c10">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#E13:8_1_1"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="tops_2.html#T46" title="TOPS_2:th.46">TOPS_2:46</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:8"/>
( ( for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Q</font> holds <br/><font color="Olive" title="b1">P</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> ) implies <font color="Maroon" title="c3">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_2"/><i><font color="Green" title="E11">A12</font></i>: 
for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Q</font> holds <br/><font color="Olive" title="b1">P</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">QQ</font> = <font color="Maroon" title="c3">Q</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="compts_1.html#E3:8"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>
<a NAME="E3:8_2"/><i><font color="Green" title="E12">A13</font></i>: 
<font color="Maroon" title="c5">QQ</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="txt" href="compts_1.html#E1:8_2"><i><font color="Green" title="E11">A12</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:8_2"/>
<font color="Maroon" title="c3">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</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>set </b><font color="Maroon" title="c7">F'</font> = <font color="Maroon" title="c6">F</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:8_2_1"/><i><font color="Green" title="E13">A14</font></i>: 
<font color="Maroon" title="c6">F</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c3">Q</font>
 <b>and </b><br/><a NAME="E2:8_2_1"/><i><font color="Green" title="E14">A15</font></i>: 
<font color="Maroon" title="c6">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:8_2_1"/><i><font color="Green" title="E15">A16</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font>
 <b>and </b><br/><a NAME="E5:8_2_1"/><i><font color="Green" title="E16">A17</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="tops_2.html#K1" title="TOPS_2:func.1">|</a> <font color="Maroon" title="c4">AA</font>
 <b>and </b><br/><a NAME="E6:8_2_1"/><i><font color="Green" title="E17">A18</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> holds <br/>for <font color="Olive" title="b2">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> holds <br/><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">AA</font>
 <b>by </b><i><a class="ref" href="tops_2.html#T50" title="TOPS_2:th.50">TOPS_2:50</a></i>;<br/>
<a NAME="E7:8_2_1"/>
<font color="Maroon" title="c6">F</font> <a href="tops_2.html#K1" title="TOPS_2:func.1">|</a> <font color="Maroon" title="c4">AA</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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">AA</font></span>)</span></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">F'</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="compts_1.html#E5:8_2_1"><i><font color="Green" title="E16">A17</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E9:8_2_1"/>
<font color="Maroon" title="c3">Q</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">F</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E1:8_2_1"><i><font color="Green" title="E13">A14</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self" title="COMPTS_1:def.1">Def1</a></i>;<br/>
<a NAME="E10:8_2_1"/><b>then </b>
<font color="Maroon" title="c5">QQ</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="c9">F'</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E4:8"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="compts_1.html#E5:8_2_1"><i><font color="Green" title="E16">A17</font></i></a>, <a class="ref" href="tops_2.html#T42" title="TOPS_2:th.42">TOPS_2:42</a></i>;<br/>
<a NAME="E11:8_2_1"/><b>then </b><i><font color="Green" title="E18">A19</font></i>: 
<font color="Maroon" title="c9">F'</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c5">QQ</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="E12:8_2_1"/>
<font color="Maroon" title="c9">F'</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:8_2_1_1"/><i><font color="Green" title="E19">A20</font></i>: 
<font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F'</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c11">Y</font> = <font color="Maroon" title="c10">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">AA</font></span>)</span> <b>by </b><i><a class="txt" href="compts_1.html#E5:8_2_1"><i><font color="Green" title="E16">A17</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">R</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:8_2_1_1"/><i><font color="Green" title="E20">A21</font></i>: 
( <font color="Maroon" title="c12">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Maroon" title="c12">R</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">AA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Y</font> )
 <b>by </b><i><a class="txt" href="compts_1.html#E5:8_2_1"><i><font color="Green" title="E16">A17</font></i></a>, <a class="txt" href="compts_1.html#E1:8_2_1_1"><i><font color="Green" title="E19">A20</font></i></a>, <a class="ref" href="tops_2.html#D3" title="TOPS_2:def.3">TOPS_2:def 3</a></i>;<br/>
<a NAME="E5:8_2_1_1"/>
<font color="Maroon" title="c12">R</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="compts_1.html#E2:8_2_1"><i><font color="Green" title="E14">A15</font></i></a>, <a class="txt" href="compts_1.html#E4:8_2_1_1"><i><font color="Green" title="E20">A21</font></i></a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></i>;<br/>
<a NAME="E6:8_2_1_1"/><b>then </b>
<font color="Maroon" title="c12">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E7:8_2_1_1"/><b>then </b>
<font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E4:8_2_1_1"><i><font color="Green" title="E20">A21</font></i></a>, <a class="ref" href="pre_topc.html#D9" title="PRE_TOPC:def.9">PRE_TOPC:def 9</a></i>;<br/>
<b>hence </b><a NAME="E8:8_2_1_1"/>
<font color="Maroon" title="c10">X</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c10">G</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E14:8_2_1"/><i><font color="Green" title="E19">A22</font></i>: 
<font color="Maroon" title="c10">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">F'</font>
 <b>and </b><br/><a NAME="E15:8_2_1"/><i><font color="Green" title="E20">A23</font></i>: 
<font color="Maroon" title="c10">G</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c5">QQ</font>
 <b>and </b><br/><a NAME="E16:8_2_1"/><i><font color="Green" title="E21">A24</font></i>: 
<font color="Maroon" title="c10">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#E3:8_2"><i><font color="Green" title="E12">A13</font></i></a>, <a class="txt" href="compts_1.html#E11:8_2_1"><i><font color="Green" title="E18">A19</font></i></a>, <a class="ref" href="compts_1.html#D7" target="_self" title="COMPTS_1:def.7">Def7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E18:8_2_1"/><i><font color="Green" title="E22">A25</font></i>: 
<font color="Maroon" title="c11">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">f</font>
 <b>and </b><br/><a NAME="E19:8_2_1"/><i><font color="Green" title="E23">A26</font></i>: 
<font color="Maroon" title="c11">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>and </b><br/><a NAME="E20:8_2_1"/><i><font color="Green" title="E24">A27</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">G</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E14:8_2_1"><i><font color="Green" title="E19">A22</font></i></a>, <a class="txt" href="compts_1.html#E16:8_2_1"><i><font color="Green" title="E21">A24</font></i></a>, <a class="ref" href="orders_1.html#T195" title="ORDERS_1:th.195">ORDERS_1:195</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">G'</font> = <font color="Maroon" title="c11">X</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="txt" href="compts_1.html#E4:8_2_1"><i><font color="Green" title="E15">A16</font></i></a>, <a class="txt" href="compts_1.html#E18:8_2_1"><i><font color="Green" title="E22">A25</font></i></a>, <a class="ref" href="tops_2.html#T3" title="TOPS_2:th.3">TOPS_2:3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">G'</font>
;<br/>

<a NAME="E22:8_2_1"/>
<font color="Maroon" title="c3">Q</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="c12">G'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:8_2_1_2"/><i><font color="Green" title="E25">A28</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font>
 ;<br/>

<a NAME="E2:8_2_1_2"/>
<font color="Maroon" title="c5">QQ</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="c10">G</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E15:8_2_1"><i><font color="Green" title="E20">A23</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self" title="COMPTS_1:def.1">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:8_2_1_2"/><i><font color="Green" title="E26">A29</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y</font>
 <b>and </b><br/><a NAME="E5:8_2_1_2"/><i><font color="Green" title="E27">A30</font></i>: 
<font color="Maroon" title="c14">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">G</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E1:8_2_1_2"><i><font color="Green" title="E25">A28</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:8_2_1_2"/><i><font color="Green" title="E28">A31</font></i>: 
<font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">f</font>
 <b>and </b><br/><a NAME="E8:8_2_1_2"/><i><font color="Green" title="E29">A32</font></i>: 
<font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font>
 <b>and </b><br/><a NAME="E9:8_2_1_2"/><i><font color="Green" title="E30">A33</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Y</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E20:8_2_1"><i><font color="Green" title="E24">A27</font></i></a>, <a class="txt" href="compts_1.html#E5:8_2_1_2"><i><font color="Green" title="E27">A30</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">Z</font> = <font color="Maroon" title="c15">z</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="compts_1.html#E4:8_2_1"><i><font color="Green" title="E15">A16</font></i></a>, <a class="txt" href="compts_1.html#E7:8_2_1_2"><i><font color="Green" title="E28">A31</font></i></a></i>;<br/>
<a NAME="E11:8_2_1_2"/>
<font color="Maroon" title="c14">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">AA</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E4:8_2_1"><i><font color="Green" title="E15">A16</font></i></a>, <a class="txt" href="compts_1.html#E6:8_2_1"><i><font color="Green" title="E17">A18</font></i></a>, <a class="txt" href="compts_1.html#E7:8_2_1_2"><i><font color="Green" title="E28">A31</font></i></a>, <a class="txt" href="compts_1.html#E9:8_2_1_2"><i><font color="Green" title="E30">A33</font></i></a></i>;<br/>
<a NAME="E12:8_2_1_2"/><b>then </b>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Z</font>
 
<b>by </b><i><a class="txt" href="compts_1.html#E4:8_2_1_2"><i><font color="Green" title="E26">A29</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:8_2_1_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c12">G'</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E8:8_2_1_2"><i><font color="Green" title="E29">A32</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E23:8_2_1"/>
( <font color="Maroon" title="c12">G'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Maroon" title="c12">G'</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c3">Q</font> &amp; <font color="Maroon" title="c12">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#E4:8_2_1"><i><font color="Green" title="E15">A16</font></i></a>, <a class="txt" href="compts_1.html#E18:8_2_1"><i><font color="Green" title="E22">A25</font></i></a>, <a class="txt" href="compts_1.html#E19:8_2_1"><i><font color="Green" title="E23">A26</font></i></a>, <a class="ref" href="compts_1.html#D1" target="_self" title="COMPTS_1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>


</div>
