<?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">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_1"/>
( <font color="Maroon" title="c2">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c2">x</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="51_1_1"><b>suppose </b></a><a NAME="E1:51_1_1"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c2">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:51_1_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_1.html#T18" title="YELLOW_1:th.18">YELLOW_1:18</a></i>;<br/><b>hence </b><a NAME="E3:51_1_1"/>
( <font color="Maroon" title="c2">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> iff <font color="Maroon" title="c2">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> )
 <b>by </b><i><a class="txt" href="waybel_8.html#E1:51_1_1"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="waybel_3.html#T15" title="WAYBEL_3:th.15">WAYBEL_3:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_2"><b>suppose </b></a><a NAME="E1:51_1_2"/><i><font color="Green" title="E25">A2</font></i>: 
not <font color="Maroon" title="c2">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:51_1_2"/>
( <font color="Maroon" title="c2">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> implies <font color="Maroon" title="c2">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1_2_1"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c2">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">Y</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/><b>assume </b><a NAME="E1:51_1_2_1_1"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">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>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $2;<br/><a NAME="E2:51_1_2_1_1"/><i><font color="Green" title="E28">A5</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> holds <br/> ex <font color="Olive" title="b2">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Olive" title="b2">u</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:51_1_2_1_1_1"/>
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font>
 ;<br/>

<a NAME="E2:51_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> )
 
<b>by </b><i><a class="txt" href="waybel_8.html#E1:51_1_2_1_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:51_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> )
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:51_1_2_1_1"/><i><font color="Green" title="E29">A6</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 <b>and </b><br/><a NAME="E5:51_1_2_1_1"/><i><font color="Green" title="E30">A7</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 <b>and </b><br/><a NAME="E6:51_1_2_1_1"/><i><font color="Green" title="E31">A8</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="waybel_8.html#E2:51_1_2_1_1"><i><font color="Green" title="E28">A5</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c5">Z</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">Y</font> <b>by </b><i><a class="txt" href="waybel_8.html#E5:51_1_2_1_1"><i><font color="Green" title="E30">A7</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">Z</font> = <font color="Maroon" title="c5">Z</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">Y</font> <b>by </b><i><a class="txt" href="waybel_8.html#E1:51_1_2_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="waybel_8.html#E4:51_1_2_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="finset_1.html#T26" title="FINSET_1:th.26">FINSET_1:26</a></i>;<br/><b>take </b><font color="Maroon" title="c7">Z</font> = <font color="Maroon" title="c6">Z</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:51_1_2_1_1_2"/><i><font color="Green" title="E32">A9</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font>
 ;<br/><a NAME="E2:51_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E33">A10</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E6:51_1_2_1_1"><i><font color="Green" title="E31">A8</font></i></a></i>;<br/><a NAME="E3:51_1_2_1_1_2"/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E4:51_1_2_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="waybel_8.html#E1:51_1_2_1_1_2"><i><font color="Green" title="E32">A9</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="E4:51_1_2_1_1_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c7">Z</font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E2:51_1_2_1_1_2"><i><font color="Green" title="E33">A10</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="E10:51_1_2_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c7">Z</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:51_1_2_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="waybel_8.html#T29" target="_self" title="WAYBEL_8:th.29">Th29</a></i>;<br/>
<b>hence </b><a NAME="E4:51_1_2_1"/>
<font color="Maroon" title="c2">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 <b>by </b><i><a class="ref" href="waybel_3.html#D2" title="WAYBEL_3:def.2">WAYBEL_3:def 2</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:51_1_2"/>
( <font color="Maroon" title="c2">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> implies <font color="Maroon" title="c2">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c3">x'</font> = <font color="Maroon" title="c2">x</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="waybel_8.html#E1:51_1_2"><i><font color="Green" title="E25">A2</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">Y</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">x'</font> : verum </span>}</span> ;<br/>
<b>assume </b><a NAME="E2:51_1_2_2"/>
<font color="Maroon" title="c2">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 ;<br/>

<a NAME="E3:51_1_2_2"/><b>then </b><i><font color="Green" title="E26">A11</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D2" title="WAYBEL_3:def.2">WAYBEL_3:def 2</a></i>;<br/>
<a NAME="E4:51_1_2_2"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">x'</font> : verum </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">t</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="E1:51_1_2_2_1"/>
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">x'</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">y'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">x'</font><b> such that </b><br/><a NAME="E3:51_1_2_2_1"/><i><font color="Green" title="E27">A12</font></i>: 
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">y'</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><a NAME="E1:51_1_2_2_1_1"/><i><font color="Green" title="E28">A13</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="waybel_8.html#T28" target="_self" title="WAYBEL_8:th.28">Th28</a></i>;<br/><b>assume </b><a NAME="E2:51_1_2_2_1_1"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">t</font>
 ;<br/><b>hence </b><a NAME="E3:51_1_2_2_1_1"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E3:51_1_2_2_1"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="waybel_8.html#E1:51_1_2_2_1_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:51_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c5">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:51_1_2_2_1"/>
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">Y</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">x'</font> : verum </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> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_2_2"><b>now </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> ;<br/><b>assume </b><a NAME="E1:51_1_2_2_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font>
 ;<br/><a NAME="E2:51_1_2_2_2"/><b>then </b><i><font color="Green" title="E27">A14</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 ;<br/><a NAME="E3:51_1_2_2_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:51_1_2_2_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E2:51_1_2_2_2"><i><font color="Green" title="E27">A14</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>
<a NAME="E7:51_1_2_2"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">Z</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c5">Y</font><b> such that </b><br/><a NAME="E9:51_1_2_2"/><i><font color="Green" title="E27">A15</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c6">Z</font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E3:51_1_2_2"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="waybel_8.html#T29" target="_self" title="WAYBEL_8:th.29">Th29</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:51_1_2_2_3"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font>
 ;<br/><a NAME="E2:51_1_2_2_3"/><b>then </b>
<font color="Maroon" title="c7">k</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="c8">y'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">x'</font><b> such that </b><br/><a NAME="E4:51_1_2_2_3"/><i><font color="Green" title="E28">A16</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">y'</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><b>thus </b><a NAME="E5:51_1_2_2_3"/>
<font color="Maroon" title="c7">k</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="waybel_8.html#E4:51_1_2_2_3"><i><font color="Green" title="E28">A16</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:51_1_2_2"/><b>then </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c6">Z</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T25" title="FINSET_1:th.25">FINSET_1:25</a></i>;<br/>
<b>hence </b><a NAME="E12:51_1_2_2"/>
<font color="Maroon" title="c2">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="waybel_8.html#E9:51_1_2_2"><i><font color="Green" title="E27">A15</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>


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

</div>
