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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>thus </b><a NAME="E1:10"/>
( <font color="Maroon">T</font> is <a href="compts_1.html#V2" target="_self">compact</a> implies for <font color="Olive">F</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>  st <font color="Olive">F</font> is <a href="compts_1.html#V1" target="_self">centered</a> &amp; <font color="Olive">F</font> is <a href="tops_2.html#V2">closed</a> holds <br/> <a href="setfam_1.html#K6">meet</a> <font color="Olive">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">T</font> is <a href="compts_1.html#V2" target="_self">compact</a>
 ;<br/>

<b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:10_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">F</font> is <a href="compts_1.html#V1" target="_self">centered</a>
 <b>and </b><br/><a NAME="E3:10_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F</font> is <a href="tops_2.html#V2">closed</a>
 ;<br/>

<b>assume </b><a NAME="E4:10_1"/><i><font color="Green">E35</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">C</font> =  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E6:10_1"/>
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, </i>;<br/>
<b>then </b> <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i><a class="ref" href="tops_2.html#T12">TOPS_2:12</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
<b>by </b><i/>
;<br/>
<a NAME="E8:10_1"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/>
<a NAME="E9:10_1"/>
 <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T16">TOPS_2:16</a></i>;<br/>
<b>then consider </b><font color="Maroon">G'</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E11:10_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">G'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E12:10_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">G'</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E13:10_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">G'</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, , </i>;<br/>
<b>set </b><font color="Maroon">F'</font> =  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">G'</font>;<br/>
<a NAME="E14:10_1"/><i><font color="Green">E40</font></i>: 
 <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">G'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:10_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">G'</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E3:10_1_2"/>
<font color="Maroon">X1</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">G'</font>
 
<b>by </b><i><a class="ref" href="compts_1.html#D2" target="_self">Def2</a>, <a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E4:10_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E5:10_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:10_1"/><i><font color="Green">E42</font></i>: 
 <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">G'</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T13">TOPS_2:13</a></i>;<br/>
<a NAME="E16:10_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">G'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T5">TOPS_2:5</a></i>;<br/>
<b>then </b><i><font color="Green">E49</font></i>:  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">G'</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">G'</font></span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i><a class="ref" href="tops_2.html#T11">TOPS_2:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span>
<b>by </b><i>, <a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>
;<br/>
<a NAME="E18:10_1"/>
 <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">G'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T10">TOPS_2:10</a></i>;<br/>
<b>hence </b><a NAME="E19:10_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="compts_1.html#D1" target="_self">Def1</a>, <a class="ref" href="compts_1.html#D2" target="_self">Def2</a>, <a class="ref" href="compts_1.html#D3" target="_self">Def3</a>, </i>;<br/>


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

<b>assume </b><a NAME="E2:10"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">F</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>  st <font color="Olive">F</font> is <a href="compts_1.html#V1" target="_self">centered</a> &amp; <font color="Olive">F</font> is <a href="tops_2.html#V2">closed</a> holds <br/> <a href="setfam_1.html#K6">meet</a> <font color="Olive">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>thus </b><a NAME="E3:10"/>
<font color="Maroon">T</font> is <a href="compts_1.html#V2" target="_self">compact</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</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="E1:10_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">F</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E2:10_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">G</font> =  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E4:10_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">G</font> is <a href="tops_2.html#V2">closed</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T17">TOPS_2:17</a></i>;<br/>
<a NAME="E5:10_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T5">TOPS_2:5</a></i>;<br/>
<b>then </b><i><font color="Green">E56</font></i>:  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">G</font> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i><a class="ref" href="tops_2.html#T11">TOPS_2:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span>
<b>by </b><i>, <a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>
;<br/>
<a NAME="E7:10_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T10">TOPS_2:10</a></i>;<br/>
<a NAME="E8:10_2"/>
not <font color="Maroon">G</font> is <a href="compts_1.html#V1" target="_self">centered</a>
 
<b>by </b><i>, , </i>;<br/>
<b>then consider </b><font color="Maroon">G'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:10_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">G'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E11:10_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">G'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E12:10_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">G'</font> is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E13:10_2"/><i><font color="Green">E64</font></i>: 
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">G'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/>
<a NAME="E14:10_2"/>
<font color="Maroon">G'</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="compts_1.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">G'</font> = <font color="Maroon">G'</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<b>take </b><font color="Maroon">F'</font> =  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">G'</font>;<br/>

<b>thus </b><a NAME="E16:10_2"/>
<font color="Maroon">F'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:10_2_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F'</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E3:10_2_2"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">G'</font>
 
<b>by </b><i><a class="ref" href="compts_1.html#D3" target="_self">Def3</a>, <a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E4:10_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="compts_1.html#D1" target="_self">Def1</a>, <a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E5:10_2_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>


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

 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F'</font> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">G'</font></span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i>, <a class="ref" href="tops_2.html#T12">TOPS_2:12</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E18:10_2"/>
<font color="Maroon">F'</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">T</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="E19:10_2"/>
<font color="Maroon">F'</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="compts_1.html#D2" target="_self">Def2</a>, <a class="ref" href="tops_2.html#T13">TOPS_2:13</a></i>;<br/>


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


</div>
