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

<b>let </b><font color="Maroon" title="c1">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">I</font> be  <a href="ami_1.html#V15" title="AMI_1:attr.15">programmed</a> <a href="ami_1.html#NM7" title="AMI_1:NM.7">FinPartState</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ;<br/>



<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = $1;<br/>
<b>set </b><font color="Maroon" title="c3">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> </span>}</span> ;<br/>
<a NAME="E1:10"/><i><font color="Green" title="E5">A1</font></i>: 
for <font color="Olive" title="b1">x</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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> holds <br/> ex <font color="Olive" title="b2">d</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">d</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_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> ;<br/>

<a NAME="E1:10_1"/><i><font color="Green" title="E6">A2</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ami_1.html#D40" title="AMI_1:def.40">AMI_1:def 40</a></i>;<br/>
<b>assume </b><a NAME="E2:10_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">l</font> = <font color="Maroon" title="c4">x</font> as    <a href="ami_1.html#M3" title="AMI_1:mode.3">Instruction-Location</a> of  <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <b>by </b><i><a class="txt" href="scmfsa6a.html#E1:10_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="ami_1.html#D4" title="AMI_1:def.4">AMI_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">d</font> = <font color="Maroon" title="c5">l</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:10_1"/>
<font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">d</font>)
 
;<br/>
<b>hence </b><a NAME="E6:10_1"/>
 ex <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">d</font>)
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:10"/><i><font color="Green" title="E6">A3</font></i>: 
for <font color="Olive" title="b1">d1</font>, <font color="Olive" title="b2">d2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">d1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">d2</font>) holds <br/><font color="Olive" title="b1">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">d2</font>
 
;<br/>
<a NAME="E3:10"/><i><font color="Green" title="E7">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font>,<span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> </span>}</span>  <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>from </b><i><a class="ref" href="funct_7.html#S3" title="FUNCT_7:sch.3">FUNCT_7:sch 3</a>(<a class="txt" href="scmfsa6a.html#E1:10"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="scmfsa6a.html#E2:10"><i><font color="Green" title="E6">A3</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><font color="Maroon">H<sub>1</sub></font>($1) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font>;<br/>
<b>set </b><font color="Maroon" title="c4">D</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">l</font>] </span>}</span> ;<br/>
<a NAME="E4:10"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">l</font>] </span>}</span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E5:10"/><b>then </b><i><font color="Green" title="E8">A5</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = $1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font>;<br/>
<b>set </b><font color="Maroon" title="c5">C</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">l</font> where <font color="Olive" title="b2">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">l</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> </span>}</span>  </span>}</span> ;<br/>
<a NAME="E6:10"/><i><font color="Green" title="E9">A6</font></i>: 
for <font color="Olive" title="b1">d1</font>, <font color="Olive" title="b2">d2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">d1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">d2</font>) holds <br/><font color="Olive" title="b1">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">d2</font>
 
;<br/>
<a NAME="E7:10"/><i><font color="Green" title="E10">A7</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> </span>}</span> ,<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">l</font> where <font color="Olive" title="b2">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">l</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> </span>}</span>  </span>}</span>  <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>from </b><i><a class="ref" href="funct_7.html#S4" title="FUNCT_7:sch.4">FUNCT_7:sch 4</a>(<a class="txt" href="scmfsa6a.html#E5:10"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="scmfsa6a.html#E6:10"><i><font color="Green" title="E9">A6</font></i></a>);<br/></i>
<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>verum;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font>;<br/>
<b>set </b><font color="Maroon" title="c6">C</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">n</font> where <font color="Olive" title="b2">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">n</font>] </span>}</span>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">l</font>] ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c7">A</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">l</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">l</font>] ) </span>}</span> ;<br/>
<a NAME="E8:10"/><i><font color="Green" title="E11">A8</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">n</font> where <font color="Olive" title="b2">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">n</font>] </span>}</span>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">l</font>] ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">l</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">l</font>] ) </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S14" title="FRAENKEL:sch.14">FRAENKEL:sch 14</a>();<br/></i>
<a NAME="E9:10"/><i><font color="Green" title="E12">A9</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">n</font> where <font color="Olive" title="b2">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">n</font>] </span>}</span>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">l</font>] ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">l</font> where <font color="Olive" title="b2">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">l</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> </span>}</span>  </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:10_2"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">n</font> where <font color="Olive" title="b2">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">n</font>] </span>}</span>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">l</font>] ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">l</font> where <font color="Olive" title="b2">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">l</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> </span>}</span>  </span>}</span> 
  <i><font color="Red">:: according to </font></i><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="10_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">e</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="E1:10_2_1"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">n</font> where <font color="Olive" title="b2">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">n</font>] </span>}</span>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">l</font>] ) </span>}</span> 
 ;<br/>

<a NAME="E2:10_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">l</font>) &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">l</font> where <font color="Olive" title="b2">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">l</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> </span>}</span>  )
 
;<br/>
<b>hence </b><a NAME="E3:10_2_1"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">l</font> where <font color="Olive" title="b2">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">l</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> </span>}</span>  </span>}</span> 
 ;<br/>


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

<b>let </b><font color="Maroon" title="c8">e</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="E2:10_2"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">l</font> where <font color="Olive" title="b2">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">l</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> </span>}</span>  </span>}</span> 
 ;<br/>

<a NAME="E3:10_2"/><b>then </b>
 ex <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">l</font> where <font color="Olive" title="b2">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">l</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> </span>}</span>  )
 
;<br/>
<b>hence </b><a NAME="E4:10_2"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">n</font> where <font color="Olive" title="b2">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">n</font>] </span>}</span>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">l</font>] ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:10"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">l</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">l</font>] ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:10_3"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">l</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">l</font>] ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><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="10_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">e</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="E1:10_3_1"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">l</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">l</font>] ) </span>}</span> 
 ;<br/>

<a NAME="E2:10_3_1"/><b>then </b>
 ex <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">l</font>) &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> )
 
;<br/>
<b>hence </b><a NAME="E3:10_3_1"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> </span>}</span> 
 ;<br/>


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

<b>let </b><font color="Maroon" title="c8">e</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="E2:10_3"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> </span>}</span> 
 ;<br/>

<a NAME="E3:10_3"/><b>then </b>
 ex <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> )
 
;<br/>
<b>hence </b><a NAME="E4:10_3"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">l</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">l</font>] ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:10"/><b>then </b>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K4" title="SCMNORM:func.4">Shift</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c1">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">l</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">l</font>] ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="valued_1.html#D12" title="VALUED_1:def.12">VALUED_1:def 12</a></i>;<br/>
<a NAME="E12:10"/><b>then </b><i><font color="Green" title="E13">A10</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K4" title="SCMNORM:func.4">Shift</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c1">m</font></span>)</span>, <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="scmfsa6a.html#E3:10"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="scmfsa6a.html#E7:10"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="scmfsa6a.html#E8:10"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="scmfsa6a.html#E9:10"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="wellord2.html#T22" title="WELLORD2:th.22">WELLORD2:22</a></i>;<br/>
<b>thus </b> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="ami_1.html#K32" title="AMI_1:func.32">ProgramPart</a> <span class="p2">(<span class="default"><a href="scmfsa_5.html#K1" title="SCMFSA_5:func.1">Relocated</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c1">m</font></span>)</span></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K3" title="SCMFSA_4:func.3">IncAddr</a> <span class="p2">(<span class="default"><a href="scmnorm.html#K4" title="SCMNORM:func.4">Shift</a> <span class="p3">(<span class="default"><a href="ami_1.html#K32" title="AMI_1:func.32">ProgramPart</a> <font color="Maroon" title="c2">I</font></span>)</span>,<font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c1">m</font></span>)</span>
<b>by </b><i><a class="ref" href="scmfsa_5.html#T2" title="SCMFSA_5:th.2">SCMFSA_5:2</a></i>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K3" title="SCMFSA_4:func.3">IncAddr</a> <span class="p2">(<span class="default"><a href="scmnorm.html#K4" title="SCMNORM:func.4">Shift</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c1">m</font></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T105" title="AMI_1:th.105">AMI_1:105</a></i>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K3" title="SCMFSA_4:func.3">IncAddr</a> <span class="p3">(<span class="default"><a href="scmnorm.html#K4" title="SCMNORM:func.4">Shift</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c1">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_circ.html#T21" title="PRE_CIRC:th.21">PRE_CIRC:21</a></i>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="scmnorm.html#K4" title="SCMNORM:func.4">Shift</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c1">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="scmfsa_4.html#D6" title="SCMFSA_4:def.6">SCMFSA_4:def 6</a></i>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa6a.html#E12:10"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">I</font>
<b>by </b><i><a class="ref" href="pre_circ.html#T21" title="PRE_CIRC:th.21">PRE_CIRC:21</a></i>
;<br/>


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