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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1"/>
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">A</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/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1"><b>suppose </b></a><a NAME="E1:39_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:39_1_1"/><b>then </b>
(  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="yellow_9.html#T16" target="_self" title="YELLOW_9:th.16">Th16</a>, <a class="ref" href="yellow_9.html#T17" target="_self" title="YELLOW_9:th.17">Th17</a></i>;<br/><a NAME="E3:39_1_1"/><b>then </b>
(  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp;  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">X</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="ref" href="yellow_9.html#T11" target="_self" title="YELLOW_9:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E4:39_1_1"/>
 <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_9.html#T18" target="_self" title="YELLOW_9:th.18">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2"><b>suppose </b></a><a NAME="E1:39_1_2"/>
<font color="Maroon" title="c2">A</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> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c2">A</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E3:39_1_2"/><i><font color="Green" title="E17">A1</font></i>: 
 <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">Y</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:39_1_2_1"/><i><font color="Green" title="E18">A2</font></i>: 
( <font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c3">A</font></span>)</span> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">Y</font> )
 <b>by </b><i><a class="ref" href="cantor_1.html#D1" title="CANTOR_1:def.1">CANTOR_1:def 1</a></i>;<br/>
<a NAME="E4:39_1_2_1"/>
<font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_1_2_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">Z</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:39_1_2_1_1"/><i><font color="Green" title="E19">A3</font></i>: 
( <font color="Maroon" title="c7">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c7">Z</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">Z</font> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E3:39_1_2_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="cantor_1.html#D4" title="CANTOR_1:def.4">CANTOR_1:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_2_1_1_1"/>
( <font color="Maroon" title="c7">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c7">Z</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/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_2_1_1_1_1"/>
<font color="Maroon" title="c7">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:39_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="yellow_9.html#E3:39_1_2_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/><a NAME="E3:39_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">A</font> &amp;  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">A</font></span>)</span> )
 <b>by </b><i><a class="ref" href="cantor_1.html#T1" title="CANTOR_1:th.1">CANTOR_1:1</a>, <a class="ref" href="cantor_1.html#T8" title="CANTOR_1:th.8">CANTOR_1:8</a></i>;<br/><b>hence </b><a NAME="E4:39_1_2_1_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">A</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_2_1_1_1_2"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c7">Z</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> 
 ;<br/><div class="add"><a NAME="E2:39_1_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c7">Z</font>
 <b>by </b><i><a class="txt" href="yellow_9.html#E3:39_1_2_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/><b>set </b><font color="Maroon" title="c8">G</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c7">Z</font>,<font color="Maroon" title="c3">A</font> : for <font color="Olive" title="b2">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">z</font> </span>}</span> ;<br/><a NAME="E3:39_1_2_1_1_1_2"/><i><font color="Green" title="E22">A6</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c7">Z</font>,<font color="Maroon" title="c3">A</font> : for <font color="Olive" title="b2">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">z</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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="E1:39_1_2_1_1_1_2_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c7">Z</font>,<font color="Maroon" title="c3">A</font> : for <font color="Olive" title="b2">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">z</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c7">Z</font>,<font color="Maroon" title="c3">A</font><b> such that </b><br/><a NAME="E3:39_1_2_1_1_1_2_1"/><i><font color="Green" title="E23">A7</font></i>: 
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">f</font></span>)</span> &amp; ( for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> holds <br/><font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">z</font> ) )
 ;<br/>
<a NAME="E4:39_1_2_1_1_1_2_1"/>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c7">Z</font>,<font color="Maroon" title="c3">A</font> is    <a href="fraenkel.html#M1" title="FRAENKEL:mode.1">FUNCTION_DOMAIN</a> of <font color="Maroon" title="c7">Z</font>,<font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="fraenkel.html#T11" title="FRAENKEL:th.11">FRAENKEL:11</a></i>;<br/>
<a NAME="E5:39_1_2_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c10">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c7">Z</font>,<font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="fraenkel.html#D2" title="FRAENKEL:def.2">FRAENKEL:def 2</a></i>;<br/>
<a NAME="E6:39_1_2_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E24">A8</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Z</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">B</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">f</font> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">B</font> = <font color="Maroon" title="c11">B</font> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E9:39_1_2_1_1_1_2_1"/>
<font color="Maroon" title="c12">B</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="yellow_9.html#E1:39_1_2_1_1_1_2"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="yellow_9.html#E6:39_1_2_1_1_1_2_1"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/>
<a NAME="E10:39_1_2_1_1_1_2_1"/><b>then </b>
(  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c12">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="txt" href="yellow_9.html#E3:39_1_2_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="yellow_9.html#E3:39_1_2_1_1_1_2_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="yellow_9.html#E6:39_1_2_1_1_1_2_1"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="finset_1.html#T26" title="FINSET_1:th.26">FINSET_1:26</a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E11:39_1_2_1_1_1_2_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="yellow_9.html#E6:39_1_2_1_1_1_2_1"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="cantor_1.html#D4" title="CANTOR_1:def.4">CANTOR_1:def 4</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c9">G</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c7">Z</font>,<font color="Maroon" title="c3">A</font> : for <font color="Olive" title="b2">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">z</font> </span>}</span>  as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">G</font> = <font color="Maroon" title="c9">G</font> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E6:39_1_2_1_1_1_2"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1_1_1_2_2"><b>proof </b></a><div class="add">

<div><b>hereby </b> <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="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon" title="c11">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_2_1_1_1_2_2_1"/>
<font color="Maroon" title="c11">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="c10">G</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c12">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_1_2_1_1_1_2_2_1"/><i><font color="Green" title="E23">A9</font></i>: 
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">b</font> &amp; <font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">G</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c7">Z</font>,<font color="Maroon" title="c3">A</font><b> such that </b><br/><a NAME="E5:39_1_2_1_1_1_2_2_1"/><i><font color="Green" title="E24">A10</font></i>: 
( <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font></span>)</span> &amp; ( for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> holds <br/><font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">z</font> ) )
 <b>by </b><i><a class="txt" href="yellow_9.html#E3:39_1_2_1_1_1_2_2_1"><i><font color="Green" title="E23">A9</font></i></a></i>;<br/><a NAME="E6:39_1_2_1_1_1_2_2_1"/>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c7">Z</font>,<font color="Maroon" title="c3">A</font> is    <a href="fraenkel.html#M1" title="FRAENKEL:mode.1">FUNCTION_DOMAIN</a> of <font color="Maroon" title="c7">Z</font>,<font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="ref" href="fraenkel.html#T11" title="FRAENKEL:th.11">FRAENKEL:11</a></i>;<br/><a NAME="E7:39_1_2_1_1_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c13">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c7">Z</font>,<font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="ref" href="fraenkel.html#D2" title="FRAENKEL:def.2">FRAENKEL:def 2</a></i>;<br/><a NAME="E8:39_1_2_1_1_1_2_2_1"/><b>then </b><i><font color="Green" title="E25">A11</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Z</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">B</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">B</font> = <font color="Maroon" title="c14">B</font> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E11:39_1_2_1_1_1_2_2_1"/>
<font color="Maroon" title="c12">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1_1_1_2_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">c</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="E1:39_1_2_1_1_1_2_2_1_1"/><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">b</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1_1_1_2_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">d</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_2_1_1_1_2_2_1_1_1"/><i><font color="Green" title="E27">A13</font></i>: 
<font color="Maroon" title="c17">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font>
 ;<br/><a NAME="E2:39_1_2_1_1_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font>
 <b>by </b><i><a class="txt" href="yellow_9.html#E8:39_1_2_1_1_1_2_2_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:39_1_2_1_1_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">d</font>
 <b>by </b><i><a class="txt" href="yellow_9.html#E5:39_1_2_1_1_1_2_2_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></i>;<br/><a NAME="E4:39_1_2_1_1_1_2_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">d</font> &amp; <font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">d</font> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E5:39_1_2_1_1_1_2_2_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="yellow_9.html#E1:39_1_2_1_1_1_2_2_1_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="yellow_9.html#E1:39_1_2_1_1_1_2_2_1_1_1"><i><font color="Green" title="E27">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E5:39_1_2_1_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">d</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:39_1_2_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="yellow_9.html#E1:39_1_2_1_1_1_2"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="yellow_9.html#E2:39_1_2_1_1_1_2"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E12:39_1_2_1_1_1_2_2_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="yellow_9.html#E3:39_1_2_1_1_1_2_2_1"><i><font color="Green" title="E23">A9</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c11">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="E2:39_1_2_1_1_1_2_2"/><i><font color="Green" title="E23">A14</font></i>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">y</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $2 &amp; $2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $1 );<br/>
<div><i><font color="Green" title="E24">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1_1_1_2_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_2_1_1_1_2_2_2"/><i><font color="Green" title="E25">A16</font></i>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font>
 ;<br/><a NAME="E2:39_1_2_1_1_1_2_2_2"/><b>then </b><i><font color="Green" title="E26">A17</font></i>: 
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">z</font> &amp; <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c3">A</font> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E3:39_1_2_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="yellow_9.html#E2:39_1_2_1_1_1_2"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="yellow_9.html#E2:39_1_2_1_1_1_2_2"><i><font color="Green" title="E23">A14</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">C</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:39_1_2_1_1_1_2_2_2"/><i><font color="Green" title="E27">A18</font></i>: 
( <font color="Maroon" title="c13">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c13">C</font> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E3:39_1_2_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="yellow_9.html#E1:39_1_2_1_1_1_2_2_2"><i><font color="Green" title="E25">A16</font></i></a>, <a class="ref" href="cantor_1.html#D1" title="CANTOR_1:def.1">CANTOR_1:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:39_1_2_1_1_1_2_2_2"/><i><font color="Green" title="E28">A19</font></i>: 
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">w</font> &amp; <font color="Maroon" title="c14">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">C</font> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E2:39_1_2_1_1_1_2_2_2"><i><font color="Green" title="E26">A17</font></i></a>, <a class="txt" href="yellow_9.html#E4:39_1_2_1_1_1_2_2_2"><i><font color="Green" title="E27">A18</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>take </b><font color="Maroon" title="c15">w</font> = <font color="Maroon" title="c14">w</font>;<br/><b>thus </b><a NAME="E7:39_1_2_1_1_1_2_2_2"/>
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="yellow_9.html#E4:39_1_2_1_1_1_2_2_2"><i><font color="Green" title="E27">A18</font></i></a>, <a class="txt" href="yellow_9.html#E6:39_1_2_1_1_1_2_2_2"><i><font color="Green" title="E28">A19</font></i></a></i>;<br/><b>thus </b><a NAME="E8:39_1_2_1_1_1_2_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">z</font>,<font color="Maroon" title="c15">w</font>]
 <b>by </b><i><a class="txt" href="yellow_9.html#E4:39_1_2_1_1_1_2_2_2"><i><font color="Green" title="E27">A18</font></i></a>, <a class="txt" href="yellow_9.html#E6:39_1_2_1_1_1_2_2_2"><i><font color="Green" title="E28">A19</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c12">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:39_1_2_1_1_1_2_2"/><i><font color="Green" title="E25">A20</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Z</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> )
 <b>and </b><br/><a NAME="E6:39_1_2_1_1_1_2_2"/><i><font color="Green" title="E26">A21</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="yellow_9.html#E3:39_1_2_1_1_1_2_2"><i><font color="Green" title="E24">A15</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c13">f</font> = <font color="Maroon" title="c12">f</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c7">Z</font>,<font color="Maroon" title="c3">A</font> <b>by </b><i><a class="txt" href="yellow_9.html#E5:39_1_2_1_1_1_2_2"><i><font color="Green" title="E25">A20</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E8:39_1_2_1_1_1_2_2"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> holds <br/><font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">z</font>
 
<b>by </b><i><a class="txt" href="yellow_9.html#E6:39_1_2_1_1_1_2_2"><i><font color="Green" title="E26">A21</font></i></a></i>;<br/>
<a NAME="E9:39_1_2_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E27">A22</font></i>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">G</font>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1_1_1_2_2_3"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:39_1_2_1_1_1_2_2_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">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="yellow_9.html#E1:39_1_2_1_1_1_2"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="yellow_9.html#E5:39_1_2_1_1_1_2_2"><i><font color="Green" title="E25">A20</font></i></a>, <a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/><b>let </b><font color="Maroon" title="c14">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E2:39_1_2_1_1_1_2_2_3"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font>
 ;<br/><a NAME="E3:39_1_2_1_1_1_2_2_3"/><b>then </b>
 ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E5:39_1_2_1_1_1_2_2"><i><font color="Green" title="E25">A20</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:39_1_2_1_1_1_2_2_3"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">y</font>
 <b>by </b><i><a class="txt" href="yellow_9.html#E6:39_1_2_1_1_1_2_2"><i><font color="Green" title="E26">A21</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:39_1_2_1_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:39_1_2_1_1_1_2_2"/>
<font color="Maroon" title="c11">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="c10">G</font>
 <b>by </b><i><a class="txt" href="yellow_9.html#E9:39_1_2_1_1_1_2_2"><i><font color="Green" title="E27">A22</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:39_1_2_1_1_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_9.html#E3:39_1_2_1_1_1_2"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="cantor_1.html#D1" title="CANTOR_1:def.1">CANTOR_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:39_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_9.html#E3:39_1_2_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="cantor_1.html#D1" title="CANTOR_1:def.1">CANTOR_1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:39_1_2"/>
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="ref" href="cantor_1.html#T1" title="CANTOR_1:th.1">CANTOR_1:1</a></i>;<br/><a NAME="E5:39_1_2"/><b>then </b>
 <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="cantor_1.html#T16" title="CANTOR_1:th.16">CANTOR_1:16</a></i>;<br/><a NAME="E6:39_1_2"/><b>then </b>
(  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> &amp;  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">A</font></span>)</span> )
 <b>by </b><i><a class="ref" href="yellow_9.html#T15" target="_self" title="YELLOW_9:th.15">Th15</a>, <a class="ref" href="cantor_1.html#T9" title="CANTOR_1:th.9">CANTOR_1:9</a></i>;<br/><b>hence </b><a NAME="E7:39_1_2"/>
 <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_9.html#E3:39_1_2"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
