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

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K3">div</a> 2</span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> );<br/>
<a NAME="E1:16_1_1"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:16_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:16_1_1_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>
<b>take </b>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
;<br/>

<b>thus </b><a NAME="E4:16_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>]
 <b>by </b><i><a class="txt" href="reloc.html#E3:16_1_1_1"><i><font color="Green">E18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:16_1_1"/><i><font color="Green">E18</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 <b>and </b><br/><a NAME="E4:16_1_1"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>(<a class="txt" href="reloc.html#E1:16_1_1"><i><font color="Green">E17</font></i></a>);<br/></i>
<a NAME="E5:16_1_1"/><i><font color="Green">E20</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="ami_3.html#K1">SCM</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></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="E1:16_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 ;<br/>

<a NAME="E2:16_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:16_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="ami_3.html#K1">SCM</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:16_1_1"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="ami_3.html#K1">SCM</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:16_1_1"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="ami_3.html#K1">SCM</a> 
 
<b>by </b><i><a class="txt" href="reloc.html#E3:16_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:16_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="ami_3.html#K1">SCM</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:16_1_1_3"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:16_1_1_3"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:16_1_1_3"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="reloc.html#E3:16_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="reloc.html#E4:16_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="ami_1.html#NM2">Instruction-Location</a> of <a href="ami_3.html#K1">SCM</a>  <b>by </b><i><a class="txt" href="reloc.html#E3:16_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="reloc.html#E5:16_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="reloc.html#E1:16_1_1_3"><i><font color="Green">E22</font></i></a></i>;<br/>
<i><font color="Green">E25</font></i>: the <a href="ami_1.html#U5">Object-Kind</a> of <a href="ami_3.html#K1">SCM</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
 <a href="ami_1.html#K3">ObjectKind</a> <font color="Maroon">c<sub>9</sub></font>

<br/>.= 
the <a href="ami_1.html#U4">Instructions</a> of <a href="ami_3.html#K1">SCM</a> 
<b>by </b><i><a class="ref" href="ami_1.html#D14">AMI_1:def 14</a></i>
;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="ami_1.html#NM4">State</a> of <a href="ami_3.html#K1">SCM</a> <b> such that </b><br/><a NAME="E8:16_1_1_3"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="ami_3.html#T39">AMI_3:39</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:16_1_1_3"/><i><font color="Green">E27</font></i>: 
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E11:16_1_1_3"/><i><font color="Green">E28</font></i>: 
 <a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="reloc.html#E3:16_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="reloc.html#E1:16_1_1_3"><i><font color="Green">E22</font></i></a>, <a class="txt" href="reloc.html#E3:16_1_1_3"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E12:16_1_1_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="reloc.html#E10:16_1_1_3"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E13:16_1_1_3"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="reloc.html#E11:16_1_1_3"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E14:16_1_1_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U4">Instructions</a> of <a href="ami_3.html#K1">SCM</a> 
 
;<br/>
<b>hence </b><a NAME="E15:16_1_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="ami_3.html#K1">SCM</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="reloc.html#E4:16_1_1_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="reloc.html#E6:16_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="reloc.html#E8:16_1_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="reloc.html#E13:16_1_1_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ami_1.html#K7">sproduct</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="ami_3.html#K1">SCM</a>  <b>by </b><i><a class="txt" href="reloc.html#E7:16_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="ami_1.html#D16">AMI_1:def 16</a></i>;<br/>
<a NAME="E10:16_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="ami_1.html#D24">AMI_1:def 24</a></i>;<br/>
<a NAME="E11:16_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T29">FINSET_1:29</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of <a href="numbers.html#K5">K94</a>() = <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K3">div</a> 2;<br/>
<a NAME="E12:16_1_1"/><i><font color="Green">E23</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>(<a class="txt" href="reloc.html#E11:16_1_1"><i><font color="Green">E22</font></i></a>);<br/></i>
<a NAME="E13:16_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K3">div</a> 2</span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_1_1_4"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K3">div</a> 2</span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></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="16_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></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="E1:16_1_1_4_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:16_1_1_4_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E4:16_1_1_4_1"/><i><font color="Green">E25</font></i>: 
 <a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 2;<br/>
<i><font color="Green">E26</font></i>: <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="nat_1.html#K3">div</a> 2
<b>by </b><i><a class="txt" href="reloc.html#E3:16_1_1_4_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="nat_1.html#T72">NAT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K3">div</a> 2
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/>
<a NAME="E6:16_1_1_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="reloc.html#E4:16_1_1_4_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:16_1_1_4_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K3">div</a> 2</span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="reloc.html#E5:16_1_1_4_1"><i><font color="Green">E26</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>8</sub></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="E2:16_1_1_4"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K3">div</a> 2</span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:16_1_1_4"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K3">div</a> 2
 <b>and </b><br/><a NAME="E5:16_1_1_4"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K3">div</a> 2;<br/>
<a NAME="E6:16_1_1_4"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="ami_3.html#K1">SCM</a> 
 
<b>by </b><i><a class="ref" href="ami_3.html#D13">AMI_3:def 13</a></i>;<br/>
<a NAME="E7:16_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K3">SCM-Instr-Loc</a> 
 
<b>by </b><i><a class="txt" href="reloc.html#E5:16_1_1_4"><i><font color="Green">E25</font></i></a>, <a class="ref" href="ami_3.html#D1">AMI_3:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:16_1_1_4"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E10:16_1_1_4"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="ami_2.html#D3">AMI_2:def 3</a></i>;<br/>
<a NAME="E11:16_1_1_4"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="reloc.html#E10:16_1_1_4"><i><font color="Green">E27</font></i></a>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<a NAME="E12:16_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E13:16_1_1_4"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="nat_1.html#K2">*</a> 1
 
<b>by </b><i><a class="txt" href="reloc.html#E9:16_1_1_4"><i><font color="Green">E26</font></i></a>, <a class="ref" href="nat_1.html#T20">NAT_1:20</a></i>;<br/>
<a NAME="E14:16_1_1_4"/>
2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="reloc.html#E9:16_1_1_4"><i><font color="Green">E26</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E15:16_1_1_4"/><b>then </b>
2 <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="txt" href="reloc.html#E13:16_1_1_4"><i><font color="Green">E28</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E16:16_1_1_4"/><b>then </b><i><font color="Green">E29</font></i>: 
2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 2
 
<b>by </b><i><a class="ref" href="nat_1.html#T57">NAT_1:57</a></i>;<br/>
<font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i><a class="txt" href="reloc.html#E13:16_1_1_4"><i><font color="Green">E28</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K3">div</a> 2</span>)</span> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i><a class="txt" href="reloc.html#E16:16_1_1_4"><i><font color="Green">E29</font></i></a>, <a class="ref" href="nat_1.html#T49">NAT_1:49</a></i>
;<br/>
<a NAME="E18:16_1_1_4"/><b>then </b>
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K3">div</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="reloc.html#E5:16_1_1_4"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E19:16_1_1_4"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="reloc.html#E4:16_1_1_4"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:16_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="reloc.html#E12:16_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="struct_0.html#M1">Element</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="ami_3.html#K1">SCM</a>  =  <a href="ami_3.html#K16">il.</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">c<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E15:16_1_1"/><i><font color="Green">E25</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>(<a class="txt" href="reloc.html#E14:16_1_1"><i><font color="Green">E24</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means  <a href="ami_3.html#K16">il.</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">c<sub>1</sub></font>;<br/>
<a NAME="E16:16_1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_1_1_5"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></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="16_1_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></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="E1:16_1_1_5_1"/>
<font color="Maroon">c<sub>8</sub></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">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:16_1_1_5_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>9</sub></font>) &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:16_1_1_5_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>10</sub></font>] )
 <b>by </b><i><a class="txt" href="reloc.html#E3:16_1_1_5_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:16_1_1_5_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="reloc.html#E3:16_1_1_5_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="reloc.html#E5:16_1_1_5_1"><i><font color="Green">E27</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>8</sub></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="E2:16_1_1_5"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:16_1_1_5"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>9</sub></font>) &amp;  <a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>
<a NAME="E5:16_1_1_5"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="reloc.html#E4:16_1_1_5"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:16_1_1_5"/>
<font color="Maroon">c<sub>8</sub></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">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  </span>}</span> 
 <b>by </b><i><a class="txt" href="reloc.html#E4:16_1_1_5"><i><font color="Green">E26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:16_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="reloc.html#E3:16_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="reloc.html#E15:16_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finset_1.html#T29">FINSET_1:29</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="ami_1.html#M1">FinPartState</a> of  <a href="ami_3.html#K1">SCM</a>  <b>by </b><i><a class="ref" href="ami_1.html#D24">AMI_1:def 24</a></i>;<br/>
<a NAME="E19:16_1_1"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="ami_3.html#V1">programmed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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>,<a class="ref" href="ami_3.html#D13">AMI_3:def 13</a><br/>

<b>assume </b><a NAME="E1:16_1_1_6"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:16_1_1_6"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="reloc.html#E3:16_1_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:16_1_1_6"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="ami_3.html#K1">SCM</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as  <a href="ami_3.html#V1">programmed</a>  <a href="ami_1.html#M1">FinPartState</a> of  <a href="ami_3.html#K1">SCM</a>  ;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E21:16_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="reloc.html#E3:16_1_1"><i><font color="Green">E18</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E22:16_1_1"/>
 <a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E23:16_1_1"/><b>then </b>
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E25:16_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E26:16_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="reloc.html#E4:16_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E27:16_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="reloc.html#E25:16_1_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E28:16_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="reloc.html#E26:16_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>


</div>
