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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> iff for <font color="Olive" title="b1">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>  st <font color="Olive" title="b1">F</font> is <a href="finset_1.html#V3" title="FINSET_1:attr.3">centered</a> &amp; <font color="Olive" title="b1">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<b>thus </b><a NAME="E1:9"/>
( <font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> implies for <font color="Olive" title="b1">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>  st <font color="Olive" title="b1">F</font> is <a href="finset_1.html#V3" title="FINSET_1:attr.3">centered</a> &amp; <font color="Olive" title="b1">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">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>  st <font color="Olive" title="b1">F</font> is <a href="finset_1.html#V3" title="FINSET_1:attr.3">centered</a> &amp; <font color="Olive" title="b1">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) implies <font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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>  st <font color="Olive" title="b1">F</font> is <a href="finset_1.html#V3" title="FINSET_1:attr.3">centered</a> &amp; <font color="Olive" title="b1">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<b>let </b><font color="Maroon" title="c2">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is <a href="finset_1.html#V3" title="FINSET_1:attr.3">centered</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> implies  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:9_1"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="finset_1.html#V3" title="FINSET_1:attr.3">centered</a>
 <b>and </b><br/><a NAME="E3:9_1"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<b>assume </b><a NAME="E4:9_1"/><i><font color="Green" title="E10">A4</font></i>: 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>reconsider </b><font color="Maroon" title="c3">C</font> =  <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c2">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> ;<br/>
<a NAME="E6:9_1"/>
<font color="Maroon" title="c2">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="#E2:9_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="finset_1.html#D3" title="FINSET_1:def.3">FINSET_1:def 3</a></i>;<br/>
<b>then </b> <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c2">F</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<b>by </b><i><a class="ref" href="tops_2.html#T12" title="TOPS_2:th.12">TOPS_2:12</a></i>
<br/>.= 
 <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="txt" href="#E4:9_1"><i><font color="Green" title="E10">A4</font></i></a></i>
;<br/>
<a NAME="E8:9_1"/><b>then </b><i><font color="Green" title="E11">A5</font></i>: 
 <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c2">F</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D12" title="SETFAM_1:def.12">SETFAM_1:def 12</a></i>;<br/>
<a NAME="E9:9_1"/>
 <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c2">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 
<b>by </b><i><a class="txt" href="#E3:9_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="tops_2.html#T16" title="TOPS_2:th.16">TOPS_2:16</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">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="E11:9_1"/><i><font color="Green" title="E12">A6</font></i>: 
<font color="Maroon" title="c4">G'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font>
 <b>and </b><br/><a NAME="E12:9_1"/><i><font color="Green" title="E13">A7</font></i>: 
<font color="Maroon" title="c4">G'</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <font color="Maroon" title="c1">T</font>
 <b>and </b><br/><a NAME="E13:9_1"/><i><font color="Green" title="E14">A8</font></i>: 
<font color="Maroon" title="c4">G'</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="#E1:9_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="#E8:9_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="compts_1.html#D3" target="_self" title="COMPTS_1:def.3">Def3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">F'</font> =  <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c4">G'</font>;<br/>
<a NAME="E14:9_1"/><i><font color="Green" title="E15">A9</font></i>: 
 <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c4">G'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c4">G'</font> or <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> )</span><br/>

<b>assume </b><a NAME="E1:9_1_2"/><i><font color="Green" title="E16">A10</font></i>: 
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c4">G'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c7">X1</font> = <font color="Maroon" title="c6">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E3:9_1_2"/>
<font color="Maroon" title="c7">X1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G'</font>
 
<b>by </b><i><a class="txt" href="#E1:9_1_2"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E4:9_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">X1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E11:9_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E5:9_1_2"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E15:9_1"/><i><font color="Green" title="E16">A11</font></i>: 
 <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c4">G'</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="#E13:9_1"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="tops_2.html#T13" title="TOPS_2:th.13">TOPS_2:13</a></i>;<br/>
<a NAME="E16:9_1"/><i><font color="Green" title="E17">A12</font></i>: 
<font color="Maroon" title="c4">G'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="#E12:9_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="tops_2.html#T5" title="TOPS_2:th.5">TOPS_2:5</a></i>;<br/>
<i><font color="Green" title="E18">A13</font></i>:  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c4">G'</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">G'</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<b>by </b><i><a class="txt" href="#E12:9_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="tops_2.html#T5" title="TOPS_2:th.5">TOPS_2:5</a>, <a class="ref" href="tops_2.html#T11" title="TOPS_2:th.11">TOPS_2:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span>
<b>by </b><i><a class="txt" href="#E12:9_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="setfam_1.html#T60" title="SETFAM_1:th.60">SETFAM_1:60</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>
;<br/>
<a NAME="E18:9_1"/>
 <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c4">G'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="#E16:9_1"><i><font color="Green" title="E17">A12</font></i></a>, <a class="ref" href="tops_2.html#T10" title="TOPS_2:th.10">TOPS_2:10</a></i>;<br/>
<b>hence </b><a NAME="E19:9_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:9_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="#E14:9_1"><i><font color="Green" title="E15">A9</font></i></a>, <a class="txt" href="#E15:9_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="txt" href="#E17:9_1"><i><font color="Green" title="E18">A13</font></i></a>, <a class="ref" href="finset_1.html#D3" title="FINSET_1:def.3">FINSET_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E2:9"/><i><font color="Green" title="E7">A14</font></i>: 
for <font color="Olive" title="b1">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>  st <font color="Olive" title="b1">F</font> is <a href="finset_1.html#V3" title="FINSET_1:attr.3">centered</a> &amp; <font color="Olive" title="b1">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a></span><br/>

<b>thus </b><a NAME="E3:9"/>
<font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">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#D3" target="_self" title="COMPTS_1:def.3">COMPTS_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c2">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> implies  ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Olive" title="b1">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:9_2"/><i><font color="Green" title="E8">A15</font></i>: 
<font color="Maroon" title="c2">F</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <font color="Maroon" title="c1">T</font>
 <b>and </b><br/><a NAME="E2:9_2"/><i><font color="Green" title="E9">A16</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Olive" title="b1">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c3">G</font> =  <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c2">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> ;<br/>
<a NAME="E4:9_2"/><i><font color="Green" title="E10">A17</font></i>: 
<font color="Maroon" title="c3">G</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a>
 
<b>by </b><i><a class="txt" href="#E2:9_2"><i><font color="Green" title="E9">A16</font></i></a>, <a class="ref" href="tops_2.html#T17" title="TOPS_2:th.17">TOPS_2:17</a></i>;<br/>
<a NAME="E5:9_2"/><i><font color="Green" title="E11">A18</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="#E1:9_2"><i><font color="Green" title="E8">A15</font></i></a>, <a class="ref" href="tops_2.html#T5" title="TOPS_2:th.5">TOPS_2:5</a></i>;<br/>
<i><font color="Green" title="E12">A19</font></i>:  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c3">G</font> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<b>by </b><i><a class="txt" href="#E1:9_2"><i><font color="Green" title="E8">A15</font></i></a>, <a class="ref" href="tops_2.html#T5" title="TOPS_2:th.5">TOPS_2:5</a>, <a class="ref" href="tops_2.html#T11" title="TOPS_2:th.11">TOPS_2:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:9_2"><i><font color="Green" title="E8">A15</font></i></a>, <a class="ref" href="setfam_1.html#T60" title="SETFAM_1:th.60">SETFAM_1:60</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>
;<br/>
<a NAME="E7:9_2"/><i><font color="Green" title="E13">A20</font></i>: 
<font color="Maroon" title="c3">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="#E5:9_2"><i><font color="Green" title="E11">A18</font></i></a>, <a class="ref" href="tops_2.html#T10" title="TOPS_2:th.10">TOPS_2:10</a></i>;<br/>
<a NAME="E8:9_2"/>
not <font color="Maroon" title="c3">G</font> is <a href="finset_1.html#V3" title="FINSET_1:attr.3">centered</a>
 
<b>by </b><i><a class="txt" href="#E2:9"><i><font color="Green" title="E7">A14</font></i></a>, <a class="txt" href="#E4:9_2"><i><font color="Green" title="E10">A17</font></i></a>, <a class="txt" href="#E6:9_2"><i><font color="Green" title="E12">A19</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">G'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:9_2"/><i><font color="Green" title="E14">A21</font></i>: 
<font color="Maroon" title="c4">G'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E11:9_2"/><i><font color="Green" title="E15">A22</font></i>: 
<font color="Maroon" title="c4">G'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G</font>
 <b>and </b><br/><a NAME="E12:9_2"/><i><font color="Green" title="E16">A23</font></i>: 
<font color="Maroon" title="c4">G'</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>and </b><br/><a NAME="E13:9_2"/><i><font color="Green" title="E17">A24</font></i>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">G'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E7:9_2"><i><font color="Green" title="E13">A20</font></i></a>, <a class="ref" href="finset_1.html#D3" title="FINSET_1:def.3">FINSET_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">G'</font> = <font color="Maroon" title="c4">G'</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="#E11:9_2"><i><font color="Green" title="E15">A22</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>take </b><font color="Maroon" title="c6">F'</font> =  <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c5">G'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">F'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c6">F'</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c6">F'</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><br/>

<b>thus </b><a NAME="E15:9_2"/>
<font color="Maroon" title="c6">F'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">F'</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c6">F'</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">A</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F'</font> or <font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> )</span><br/>

<b>assume </b><a NAME="E1:9_2_2"/><i><font color="Green" title="E18">A25</font></i>: 
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c8">A1</font> = <font color="Maroon" title="c7">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/>
<a NAME="E3:9_2_2"/>
<font color="Maroon" title="c8">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G'</font>
 
<b>by </b><i><a class="txt" href="#E1:9_2_2"><i><font color="Green" title="E18">A25</font></i></a>, <a class="ref" href="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E4:9_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E11:9_2"><i><font color="Green" title="E15">A22</font></i></a>, <a class="ref" href="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E5:9_2_2"/>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c6">F'</font> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">G'</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<b>by </b><i><a class="txt" href="#E10:9_2"><i><font color="Green" title="E14">A21</font></i></a>, <a class="ref" href="tops_2.html#T12" title="TOPS_2:th.12">TOPS_2:12</a></i>
<br/>.= 
 <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="txt" href="#E13:9_2"><i><font color="Green" title="E17">A24</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E17:9_2"/>
<font color="Maroon" title="c6">F'</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D12" title="SETFAM_1:def.12">SETFAM_1:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">F'</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a></span><br/>

<b>thus </b><a NAME="E18:9_2"/>
<font color="Maroon" title="c6">F'</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="#E12:9_2"><i><font color="Green" title="E16">A23</font></i></a>, <a class="ref" href="tops_2.html#T13" title="TOPS_2:th.13">TOPS_2:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div>
