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

<b>let </b><font color="Maroon">L</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">L</font>, <font color="Maroon">G1</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:81"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">G</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E2:81"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>set </b><font color="Maroon">ZAW</font> = <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> ;<br/>
<b>let </b><font color="Maroon">ALL</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:81"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:81"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">ALL</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">L</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> ) </span>}</span> 
 ;<br/>

<b>set </b><font color="Maroon">R</font> =  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">ALL</font>;<br/>
<a NAME="E5:81"/>
<font color="Maroon">G1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T14" target="_self">Th14</a>, <a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E6:81"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">G1</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E7:81"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">G1</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T7" target="_self">Th7</a>, <a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_3.html#T14" target="_self">Th14</a>, </i>;<br/>
<a NAME="E8:81"/><i><font color="Green">E48</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">ALL</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">ALL</font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E9:81"/><i><font color="Green">E49</font></i>: 
 <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">ALL</font> is <a href="relat_2.html#V1">reflexive</a>
 
<b>by </b><i><a class="ref" href="wellord2.html#T2">WELLORD2:2</a></i>;<br/>
<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D10">ORDERS_1:def 10</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E10:81"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">ALL</font>
 <b>and </b><br/><a NAME="E11:81"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">ALL</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font> <a href="orders_1.html#NR3" target="_self">is_linear-order</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_1"/>
(  not <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1"><b>suppose </b></a><a NAME="E1:81_1_1"/><i><font color="Green">E52</font></i>: 
not <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>take </b><font color="Maroon">x</font> =  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Y</font>;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:81_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T21" target="_self">Th21</a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:81_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ALL</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="rcomp_3.html#T22" target="_self">Th22</a></i>;<br/><b>then consider </b><font color="Maroon">C</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:81_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E7:81_1_1"/>
<font color="Maroon">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E8:81_1_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E9:81_1_1"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T22" target="_self">Th22</a>, <a class="ref" href="rcomp_3.html#T23" target="_self">Th23</a>, <a class="ref" href="setfam_1.html#T8">SETFAM_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">L</font> ;<br/><a NAME="E11:81_1_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T23" target="_self">Th23</a>, <a class="ref" href="rcomp_3.html#T24" target="_self">Th24</a>, <a class="ref" href="rcomp_3.html#T22" target="_self">Th22</a>, <a class="ref" href="setfam_1.html#T8">SETFAM_1:8</a></i>;<br/><a NAME="E12:81_1_1"/><i><font color="Green">E97</font></i>: 
 <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">G1</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="finset_1.html#T24">FINSET_1:24</a></i>;<br/><b>set </b><font color="Maroon">E</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">D</font>) where <font color="Olive">D</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">L</font> : <font color="Olive">D</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">G1</font> </span>}</span> ;<br/><a NAME="E13:81_1_1"/><i><font color="Green">E98</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">D</font>) where <font color="Olive">D</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">L</font> : <font color="Olive">D</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">G1</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>(<a class="ref" href="rcomp_3.html#T26" target="_self">Th26</a>);<br/></i><a NAME="E14:81_1_1"/>
<font color="Maroon">ALL</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">D</font>) where <font color="Olive">D</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">L</font> : <font color="Olive">D</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">G1</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_1"><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="E1:81_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ALL</font>
 ;<br/>

<a NAME="E2:81_1_1_1"/><b>then </b>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">L</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> )
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T16" target="_self">Th16</a></i>;<br/>
<b>hence </b><a NAME="E3:81_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">D</font>) where <font color="Olive">D</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">L</font> : <font color="Olive">D</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">G1</font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div><a NAME="E15:81_1_1"/><b>then </b>
<font color="Maroon">ALL</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E16:81_1_1"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">Y</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( not <font color="Maroon">a<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> implies  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> );<br/><a NAME="E17:81_1_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 ;<br/><a NAME="E18:81_1_1"/><i><font color="Green">E101</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_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> , <font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:81_1_1_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E2:81_1_1_2"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E3:81_1_1_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">B</font>]
 <b>and </b><br/><a NAME="E4:81_1_1_2"/>
not <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_1_1_2_1"/>
( <font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:81_1_1_2_1_1"/>
<font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:81_1_1_2_1_1"/>
 <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a>, <a class="ref" href="setfam_1.html#T11">SETFAM_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:81_1_1_2_1_2"/><i><font color="Green">E104</font></i>: 
not <font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then </b><i><font color="Green">E105</font></i>:  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">B</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="setfam_1.html#T10">SETFAM_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">B</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="setfam_1.html#T11">SETFAM_1:11</a></i>
;<br/><a NAME="E3:81_1_1_2_1_2"/><i><font color="Green">E106</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">ALL</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T19" target="_self">Th19</a>, <a class="ref" href="rcomp_3.html#T20" target="_self">Th20</a>, <a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="orders_1.html#T181">ORDERS_1:181</a></i>;<br/><a NAME="E4:81_1_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">ALL</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font> is <a href="relat_2.html#V6">connected</a>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E23"><i><font color="Green">Lemma66</font></i></a>, <a class="ref" href="orders_1.html#D5">ORDERS_1:def 5</a></i>;<br/><a NAME="E5:81_1_1_2_1_2"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">ALL</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">ALL</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_2.html#D14">RELAT_2:def 14</a></i>;<br/><a NAME="E6:81_1_1_2_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">B</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">ALL</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">ALL</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a>, <a class="ref" href="rcomp_3.html#T32" target="_self">Th32</a>, <a class="ref" href="rcomp_3.html#T33" target="_self">Th33</a>, <a class="ref" href="rcomp_3.html#T35" target="_self">Th35</a>, <a class="ref" href="rcomp_3.html#T36" target="_self">Th36</a>, <a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a></i>;<br/><a NAME="E7:81_1_1_2_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">B</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">ALL</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">ALL</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:81_1_1_2_1_2"/><b>then </b>
(  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> or <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a>, <a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="rcomp_3.html#T32" target="_self">Th32</a>, <a class="ref" href="rcomp_3.html#T33" target="_self">Th33</a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/><b>hence </b><a NAME="E9:81_1_1_2_1_2"/>
 <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a>, <a class="ref" href="rcomp_3.html#T34" target="_self">Th34</a>, <a class="ref" href="rcomp_3.html#T32" target="_self">Th32</a>, <a class="ref" href="rcomp_3.html#T33" target="_self">Th33</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E19:81_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">Y</font>]
 <b>from </b><i><a class="ref" href="finset_1.html#S2">FINSET_1:sch 2</a>(<a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a>, <a class="ref" href="rcomp_3.html#T29" target="_self">Th29</a>, <a class="ref" href="rcomp_3.html#T30" target="_self">Th30</a>);<br/></i><a NAME="E20:81_1_1"/>
<font color="Maroon">X</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:81_1_1_3"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_3_1"><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="E1:81_1_1_3_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font>
 ;<br/>

<a NAME="E2:81_1_1_3_1"/>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ALL</font>
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a>, <a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="rcomp_3.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E3:81_1_1_3_1"/><b>then </b>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">L</font> st <br/>(  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">L</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> )
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T16" target="_self">Th16</a></i>;<br/>
<b>hence </b><a NAME="E4:81_1_1_3_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T32" target="_self">Th32</a>, <a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/>


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

<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="E2:81_1_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:81_1_1_3"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E5:81_1_1_3"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:81_1_1_3"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T22" target="_self">Th22</a>, <a class="ref" href="rcomp_3.html#T33" target="_self">Th33</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>thus </b><a NAME="E7:81_1_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T32" target="_self">Th32</a>, <a class="ref" href="rcomp_3.html#T23" target="_self">Th23</a>, <a class="ref" href="rcomp_3.html#T34" target="_self">Th34</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E21:81_1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ALL</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T25" target="_self">Th25</a>, <a class="ref" href="rcomp_3.html#T16" target="_self">Th16</a></i>;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E22:81_1_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E23:81_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T8">SETFAM_1:8</a></i>;<br/><b>hence </b><a NAME="E24:81_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">ALL</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="rcomp_3.html#T32" target="_self">Th32</a>, <a class="ref" href="rcomp_3.html#T33" target="_self">Th33</a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_2"><b>suppose </b></a><a NAME="E1:81_1_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">G1</font>
;<br/><b>thus </b><a NAME="E2:81_1_2"/>
( <font color="Maroon">G1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ALL</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">G1</font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">ALL</font> ) ) )
 <b>by </b><i><a class="ref" href="rcomp_3.html#T18" target="_self">Th18</a>, <a class="ref" href="rcomp_3.html#T21" target="_self">Th21</a>, <a class="ref" href="rcomp_3.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
