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

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">I</font> be  <a href="ami_3.html#V1">programmed</a>  <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/>



<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a>  =  <a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span> ;<br/>
<a NAME="E1:18"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> holds <br/> ex <font color="Olive">d</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">d</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:18_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 
<b>by </b><i><a class="ref" href="ami_3.html#D13">AMI_3:def 13</a></i>;<br/>
<b>hence </b><a NAME="E2:18_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> implies  ex <font color="Olive">d</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">d</font>) )
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T21">SCMFSA_2:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:18"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">d1</font>, <font color="Olive">d2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">d1</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">d2</font>) holds <br/><font color="Olive">d1</font> <a href="hidden.html#R1">=</a> <font color="Olive">d2</font>
 
<b>by </b><i><a class="ref" href="scmfsa_2.html#T18">SCMFSA_2:18</a></i>;<br/>
<a NAME="E3:18"/><i><font color="Green">E31</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font>,<span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span>  <a href="tarski.html#R2">are_equipotent</a> 
 
<b>from </b><i><a class="ref" href="funct_7.html#S3">FUNCT_7:sch 3</a>(, );<br/></i>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font>;<br/>
<b>set </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l</font>] </span>}</span> ;<br/>
<a NAME="E4:18"/>
<span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l</font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> 
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E5:18"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a>  =  <a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">C</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span>  </span>}</span> ;<br/>
<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">d1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">d2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:18_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">d1</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">d2</font>)
 ;<br/><a NAME="E2:18_2"/><b>then </b>
<font color="Maroon">d1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T18">SCMFSA_2:18</a></i>;<br/><b>hence </b><a NAME="E3:18_2"/>
<font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d2</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E7:18"/><i><font color="Green">E34</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span> ,<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span>  </span>}</span>  <a href="tarski.html#R2">are_equipotent</a> 
 
<b>from </b><i><a class="ref" href="funct_7.html#S4">FUNCT_7:sch 4</a>(, );<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>verum;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> <a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font>;<br/>
<b>set </b><font color="Maroon">C</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">n</font> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] </span>}</span>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">l</font>] ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">l</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">l</font>] ) </span>}</span> ;<br/>
<a NAME="E8:18"/><i><font color="Green">E38</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">n</font> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] </span>}</span>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">l</font>] ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">l</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">l</font>] ) </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S14">FRAENKEL:sch 14</a>();<br/></i>
<a NAME="E9:18"/><i><font color="Green">E40</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">n</font> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] </span>}</span>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">l</font>] ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span>  </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:18_3"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">n</font> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] </span>}</span>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">l</font>] ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span>  </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:18_3_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">n</font> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] </span>}</span>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">l</font>] ) </span>}</span> 
 ;<br/>

<a NAME="E2:18_3_1"/><b>then </b>
 ex <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>) &amp; <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span>  )
 
;<br/>
<b>hence </b><a NAME="E3:18_3_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span>  </span>}</span> 
 ;<br/>


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

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

<b>assume </b><a NAME="E2:18_3"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span>  </span>}</span> 
 ;<br/>

<a NAME="E3:18_3"/><b>then </b>
 ex <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span>  )
 
;<br/>
<b>hence </b><a NAME="E4:18_3"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">n</font> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] </span>}</span>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">l</font>] ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:18"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">l</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">l</font>] ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:18_4"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">l</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">l</font>] ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:18_4_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">l</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">l</font>] ) </span>}</span> 
 ;<br/>

<a NAME="E2:18_4_1"/><b>then </b>
 ex <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>) &amp;  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> )
 
;<br/>
<b>hence </b><a NAME="E3:18_4_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span> 
 ;<br/>


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

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

<b>assume </b><a NAME="E2:18_4"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span> 
 ;<br/>

<a NAME="E3:18_4"/><b>then </b>
 ex <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> &amp;  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> )
 
;<br/>
<b>hence </b><a NAME="E4:18_4"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">l</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">l</font>] ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:18"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K8">Shift</a> <font color="Maroon">I</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">l</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">l</font>] ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="scmfsa_4.html#D7">SCMFSA_4:def 7</a></i>;<br/>
<a NAME="E12:18"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K8">Shift</a> <font color="Maroon">I</font>,<font color="Maroon">m</font></span>)</span>, <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i>, , , , <a class="ref" href="wellord2.html#T22">WELLORD2:22</a></i>;<br/>
<b>thus </b> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="ami_5.html#K6">ProgramPart</a> <span class="p2">(<span class="default"><a href="scmfsa_5.html#K1">Relocated</a> <font color="Maroon">I</font>,<font color="Maroon">m</font></span>)</span></span>)</span> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K7">IncAddr</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K8">Shift</a> <span class="p3">(<span class="default"><a href="ami_5.html#K6">ProgramPart</a> <font color="Maroon">I</font></span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="scmfsa_5.html#T2">SCMFSA_5:2</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K7">IncAddr</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K8">Shift</a> <font color="Maroon">I</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="ami_5.html#T72">AMI_5:72</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K7">IncAddr</a> <span class="p3">(<span class="default"><a href="scmfsa_4.html#K8">Shift</a> <font color="Maroon">I</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K8">Shift</a> <font color="Maroon">I</font>,<font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="scmfsa_4.html#D6">SCMFSA_4:def 6</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font></span>)</span>
<b>by </b><i>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>
;<br/>


</div>
