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

<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c3">G1</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:72"/><i><font color="Green" title="E55">A1</font></i>: 
<font color="Maroon" title="c2">G</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">L</font>
 <b>and </b><br/><a NAME="E2:72"/><i><font color="Green" title="E56">A2</font></i>: 
<font color="Maroon" title="c2">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">ZAW</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> &amp;  ex <font color="Olive" title="b2">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">X</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b2">Y</font> ) ) </span>}</span> ;<br/>
<b>let </b><font color="Maroon" title="c5">ALL</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:72"/><i><font color="Green" title="E57">A3</font></i>: 
<font color="Maroon" title="c3">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> &amp;  ex <font color="Olive" title="b2">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">X</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b2">Y</font> ) ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:72"/><i><font color="Green" title="E58">A4</font></i>: 
<font color="Maroon" title="c5">ALL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">C</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Olive" title="b1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G1</font> ) </span>}</span> 
 ;<br/>

<b>set </b><font color="Maroon" title="c6">R</font> =  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c5">ALL</font>;<br/>
<a NAME="E5:72"/><i><font color="Green" title="E59">A5</font></i>: 
<font color="Maroon" title="c3">G1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E2:72"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="rcomp_3.html#E3:72"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E6:72"/><i><font color="Green" title="E60">A6</font></i>: 
<font color="Maroon" title="c3">G1</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E1:72"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="rcomp_3.html#E2:72"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="rcomp_3.html#E3:72"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="rcomp_3.html#T59" target="_self" title="RCOMP_3:th.59">Th59</a></i>;<br/>
<a NAME="E7:72"/><i><font color="Green" title="E61">A7</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c5">ALL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">ALL</font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">Y</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="orders_1.html#D10" title="ORDERS_1:def.10">ORDERS_1:def 10</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E8:72"/><i><font color="Green" title="E62">A8</font></i>: 
<font color="Maroon" title="c7">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">ALL</font>
 <b>and </b><br/><a NAME="E9:72"/><i><font color="Green" title="E63">A9</font></i>: 
<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c5">ALL</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c7">Y</font> <a href="orders_1.html#NR3" title="ORDERS_1:NR.3">is_linear-order</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:72_1"/>
(  not <font color="Maroon" title="c7">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or <font color="Maroon" title="c7">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1"><b>suppose </b></a><a NAME="E1:72_1_1"/><i><font color="Green" title="E64">A10</font></i>: 
not <font color="Maroon" title="c7">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c8">x</font> =  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c7">Y</font>;<br/><b>consider </b><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:72_1_1"/><i><font color="Green" title="E65">A11</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:72_1_1"><i><font color="Green" title="E64">A10</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:72_1_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ALL</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E8:72"><i><font color="Green" title="E62">A8</font></i></a>, <a class="txt" href="rcomp_3.html#E3:72_1_1"><i><font color="Green" title="E65">A11</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">C</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E6:72_1_1"/><i><font color="Green" title="E66">A12</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C</font>
 <b>and </b><br/><a NAME="E7:72_1_1"/>
<font color="Maroon" title="c10">C</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">L</font>
 <b>and </b><br/><a NAME="E8:72_1_1"/><i><font color="Green" title="E67">A13</font></i>: 
<font color="Maroon" title="c10">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G1</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E4:72"><i><font color="Green" title="E58">A4</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">X</font> = <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="rcomp_3.html#E3:72_1_1"><i><font color="Green" title="E65">A11</font></i></a>, <a class="txt" href="rcomp_3.html#E6:72_1_1"><i><font color="Green" title="E66">A12</font></i></a>, <a class="ref" href="setfam_1.html#T8" title="SETFAM_1:th.8">SETFAM_1:8</a></i>;<br/><a NAME="E10:72_1_1"/><i><font color="Green" title="E68">A14</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G1</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E3:72_1_1"><i><font color="Green" title="E65">A11</font></i></a>, <a class="txt" href="rcomp_3.html#E6:72_1_1"><i><font color="Green" title="E66">A12</font></i></a>, <a class="txt" href="rcomp_3.html#E8:72_1_1"><i><font color="Green" title="E67">A13</font></i></a>, <a class="ref" href="setfam_1.html#T8" title="SETFAM_1:th.8">SETFAM_1:8</a></i>;<br/><a NAME="E11:72_1_1"/><i><font color="Green" title="E69">A15</font></i>: 
 <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c3">G1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E5:72"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="finset_1.html#T24" title="FINSET_1:th.24">FINSET_1:24</a></i>;<br/><b>set </b><font color="Maroon" title="c12">E</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">D</font>) where <font color="Olive" title="b1">D</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c3">G1</font> </span>}</span> ;<br/><a NAME="E12:72_1_1"/><i><font color="Green" title="E70">A16</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">D</font>) where <font color="Olive" title="b1">D</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c3">G1</font> </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21" title="FRAENKEL:sch.21">FRAENKEL:sch 21</a>(<a class="txt" href="rcomp_3.html#E11:72_1_1"><i><font color="Green" title="E69">A15</font></i></a>);<br/></i><a NAME="E13:72_1_1"/>
<font color="Maroon" title="c5">ALL</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">D</font>) where <font color="Olive" title="b1">D</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c3">G1</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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><br/>

<b>assume </b><a NAME="E1:72_1_1_1"/>
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ALL</font>
 ;<br/>

<a NAME="E2:72_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b1">C</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Olive" title="b1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G1</font> )
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E4:72"><i><font color="Green" title="E58">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:72_1_1_1"/>
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">D</font>) where <font color="Olive" title="b1">D</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c3">G1</font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div><a NAME="E14:72_1_1"/><b>then </b>
<font color="Maroon" title="c5">ALL</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E12:72_1_1"><i><font color="Green" title="E70">A16</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/><a NAME="E15:72_1_1"/><b>then </b><i><font color="Green" title="E71">A17</font></i>: 
<font color="Maroon" title="c7">Y</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E8:72"><i><font color="Green" title="E62">A8</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( not $1 is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> );<br/><a NAME="E16:72_1_1"/><i><font color="Green" title="E72">A18</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 ;<br/><a NAME="E17:72_1_1"/><i><font color="Green" title="E73">A19</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">B</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="c7">Y</font> &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Y</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">B</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:72_1_1_2"/><i><font color="Green" title="E74">A20</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font>
 <b>and </b><br/><a NAME="E2:72_1_1_2"/>
<font color="Maroon" title="c14">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Y</font>
 <b>and </b><br/><a NAME="E3:72_1_1_2"/><i><font color="Green" title="E75">A21</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c14">B</font>]
 <b>and </b><br/><a NAME="E4:72_1_1_2"/>
not <font color="Maroon" title="c14">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_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="E1:72_1_1_2_1"/>
( <font color="Maroon" title="c14">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c14">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:72_1_1_2_1_1"/>
<font color="Maroon" title="c14">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:72_1_1_2_1_1"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:72_1_1_2"><i><font color="Green" title="E74">A20</font></i></a>, <a class="ref" href="setfam_1.html#T11" title="SETFAM_1:th.11">SETFAM_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:72_1_1_2_1_2"/><i><font color="Green" title="E76">A22</font></i>: 
not <font color="Maroon" title="c14">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E77">A23</font></i>:  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c14">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="setfam_1.html#T10" title="SETFAM_1:th.10">SETFAM_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c14">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="ref" href="setfam_1.html#T11" title="SETFAM_1:th.11">SETFAM_1:11</a></i>
;<br/><a NAME="E3:72_1_1_2_1_2"/><i><font color="Green" title="E78">A24</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c5">ALL</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c7">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Y</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E7:72"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="rcomp_3.html#E8:72"><i><font color="Green" title="E62">A8</font></i></a>, <a class="ref" href="orders_1.html#T181" title="ORDERS_1:th.181">ORDERS_1:181</a>, <a class="ref" href="wellord2.html#T2" title="WELLORD2:th.2">WELLORD2:2</a></i>;<br/><a NAME="E4:72_1_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c5">ALL</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c7">Y</font> is <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E9:72"><i><font color="Green" title="E63">A9</font></i></a>, <a class="ref" href="orders_1.html#D5" title="ORDERS_1:def.5">ORDERS_1:def 5</a></i>;<br/><a NAME="E5:72_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c5">ALL</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c7">Y</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c5">ALL</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c7">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_2.html#D14" title="RELAT_2:def.14">RELAT_2:def 14</a></i>;<br/><a NAME="E6:72_1_1_2_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c14">B</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c5">ALL</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c7">Y</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c14">B</font></span>)</span>,<font color="Maroon" title="c13">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c5">ALL</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c7">Y</font> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c14">B</font> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:72_1_1_2"><i><font color="Green" title="E74">A20</font></i></a>, <a class="txt" href="rcomp_3.html#E3:72_1_1_2"><i><font color="Green" title="E75">A21</font></i></a>, <a class="txt" href="rcomp_3.html#E1:72_1_1_2_1_2"><i><font color="Green" title="E76">A22</font></i></a>, <a class="txt" href="rcomp_3.html#E3:72_1_1_2_1_2"><i><font color="Green" title="E78">A24</font></i></a>, <a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a></i>;<br/><a NAME="E7:72_1_1_2_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c14">B</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c5">ALL</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c14">B</font></span>)</span>,<font color="Maroon" title="c13">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c5">ALL</font> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c14">B</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:72_1_1_2_1_2"/><b>then </b>
(  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c14">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">x</font> or <font color="Maroon" title="c13">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c14">B</font> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E8:72"><i><font color="Green" title="E62">A8</font></i></a>, <a class="txt" href="rcomp_3.html#E1:72_1_1_2"><i><font color="Green" title="E74">A20</font></i></a>, <a class="txt" href="rcomp_3.html#E3:72_1_1_2"><i><font color="Green" title="E75">A21</font></i></a>, <a class="txt" href="rcomp_3.html#E1:72_1_1_2_1_2"><i><font color="Green" title="E76">A22</font></i></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/><b>hence </b><a NAME="E9:72_1_1_2_1_2"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:72_1_1_2"><i><font color="Green" title="E74">A20</font></i></a>, <a class="txt" href="rcomp_3.html#E3:72_1_1_2"><i><font color="Green" title="E75">A21</font></i></a>, <a class="txt" href="rcomp_3.html#E1:72_1_1_2_1_2"><i><font color="Green" title="E76">A22</font></i></a>, <a class="txt" href="rcomp_3.html#E2:72_1_1_2_1_2"><i><font color="Green" title="E77">A23</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E18:72_1_1"/><i><font color="Green" title="E74">A25</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">Y</font>]
 <b>from </b><i><a class="ref" href="finset_1.html#S2" title="FINSET_1:sch.2">FINSET_1:sch 2</a>(<a class="txt" href="rcomp_3.html#E15:72_1_1"><i><font color="Green" title="E71">A17</font></i></a>, <a class="txt" href="rcomp_3.html#E16:72_1_1"><i><font color="Green" title="E72">A18</font></i></a>, <a class="txt" href="rcomp_3.html#E17:72_1_1"><i><font color="Green" title="E73">A19</font></i></a>);<br/></i><a NAME="E19:72_1_1"/>
<font color="Maroon" title="c11">X</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:72_1_1_3"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</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="c11">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D8" title="PRE_TOPC:def.8">PRE_TOPC:def 8</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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><br/>

<b>assume </b><a NAME="E1:72_1_1_3_1"/><i><font color="Green" title="E75">A26</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font>
 ;<br/>

<a NAME="E2:72_1_1_3_1"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ALL</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E8:72"><i><font color="Green" title="E62">A8</font></i></a>, <a class="txt" href="rcomp_3.html#E1:72_1_1"><i><font color="Green" title="E64">A10</font></i></a>, <a class="txt" href="rcomp_3.html#E18:72_1_1"><i><font color="Green" title="E74">A25</font></i></a></i>;<br/>
<a NAME="E3:72_1_1_3_1"/><b>then </b>
 ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font> st <br/>(  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b1">C</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Olive" title="b1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G1</font> )
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E4:72"><i><font color="Green" title="E58">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:72_1_1_3_1"/>
<font color="Maroon" title="c13">a</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="c11">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:72_1_1_3_1"><i><font color="Green" title="E75">A26</font></i></a>, <a class="ref" href="pre_topc.html#D8" title="PRE_TOPC:def.8">PRE_TOPC:def 8</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c13">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><br/>

<b>assume </b><a NAME="E2:72_1_1_3"/>
<font color="Maroon" title="c13">a</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="c11">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:72_1_1_3"/><i><font color="Green" title="E75">A27</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Z</font>
 <b>and </b><br/><a NAME="E5:72_1_1_3"/><i><font color="Green" title="E76">A28</font></i>: 
<font color="Maroon" title="c14">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>thus </b><a NAME="E6:72_1_1_3"/>
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E4:72_1_1_3"><i><font color="Green" title="E75">A27</font></i></a>, <a class="txt" href="rcomp_3.html#E5:72_1_1_3"><i><font color="Green" title="E76">A28</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E20:72_1_1"/><i><font color="Green" title="E75">A29</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ALL</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E4:72"><i><font color="Green" title="E58">A4</font></i></a>, <a class="txt" href="rcomp_3.html#E10:72_1_1"><i><font color="Green" title="E68">A14</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E21:72_1_1"/><i><font color="Green" title="E76">A30</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font>
 ;<br/><a NAME="E22:72_1_1"/><b>then </b>
<font color="Maroon" title="c8">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">y</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T8" title="SETFAM_1:th.8">SETFAM_1:8</a></i>;<br/><b>hence </b><a NAME="E23:72_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c13">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c5">ALL</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E8:72"><i><font color="Green" title="E62">A8</font></i></a>, <a class="txt" href="rcomp_3.html#E20:72_1_1"><i><font color="Green" title="E75">A29</font></i></a>, <a class="txt" href="rcomp_3.html#E21:72_1_1"><i><font color="Green" title="E76">A30</font></i></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_2"><b>suppose </b></a><a NAME="E1:72_1_2"/><i><font color="Green" title="E64">A31</font></i>: 
<font color="Maroon" title="c7">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c3">G1</font>
;<br/><b>thus </b><a NAME="E2:72_1_2"/>
( <font color="Maroon" title="c3">G1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ALL</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">G1</font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c5">ALL</font> ) ) )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E4:72"><i><font color="Green" title="E58">A4</font></i></a>, <a class="txt" href="rcomp_3.html#E6:72"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="rcomp_3.html#E1:72_1_2"><i><font color="Green" title="E64">A31</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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