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<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> ] <b>means </b> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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> <font color="Olive">m</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="reloc.html#K3" target="_self">IncAddr</a> <span class="p1">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="ami_3.html#K16">il.</a> <font color="Olive">m</font></span>)</span></span>)</span>,<font color="Maroon">k</font> );<br/>
<a NAME="E1:20_1_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:20_1_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:20_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</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/>
<b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:20_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="ami_5.html#T19">AMI_5:19</a></i>;<br/>
<b>take </b>
 <a href="reloc.html#K3" target="_self">IncAddr</a> <span class="p1">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="ami_3.html#K16">il.</a> <font color="Maroon">m</font></span>)</span></span>)</span>,<font color="Maroon">k</font>
;<br/>

<b>thus </b><a NAME="E5:20_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">e</font>, <a href="reloc.html#K3" target="_self">IncAddr</a> <span class="p1">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="ami_3.html#K16">il.</a> <font color="Maroon">m</font></span>)</span></span>)</span>,<font color="Maroon">k</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:20_1_1"/><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E4:20_1_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<a NAME="E5:20_1_1"/><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</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="E6:20_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <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="E7:20_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</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="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:20_1_1"/>
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">f</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</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">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:20_1_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:20_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E4:20_1_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="reloc.html#K3" target="_self">IncAddr</a> <span class="p1">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="ami_3.html#K16">il.</a> <font color="Maroon">m</font></span>)</span></span>)</span>,<font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</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#E1:20_1_1"><i><font color="Green">E24</font></i></a></i>;<br/>
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">y</font> = 
 <a href="ami_1.html#K3">ObjectKind</a> <font color="Maroon">y</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>hence </b><a NAME="E7:20_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</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="ref" href="reloc.html#D1" target="_self">Def1</a>, <a class="ref" href="ami_1.html#D16">AMI_1:def 16</a></i>;<br/>
<a NAME="E10:20_1_1"/>
<font color="Maroon">p</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="E11:20_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</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/>
<a NAME="E12:20_1_1"/><b>then </b>
<font color="Maroon">f</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>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</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="E14:20_1_1"/>
<font color="Maroon">f</font> is <a href="ami_3.html#V1">programmed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:20_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>hence </b><a NAME="E2:20_1_1_3"/>
<font color="Maroon">x</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> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">IT</font> = <font color="Maroon">f</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">IT</font>
;<br/>

<b>thus </b><a NAME="E16:20_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>

<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>assume </b><a NAME="E17:20_1_1"/>
 <a href="ami_3.html#K16">il.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E19:20_1_1"/><i><font color="Green">E67</font></i>: 
 <a href="ami_3.html#K16">il.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E20:20_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">f</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">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="reloc.html#K3" target="_self">IncAddr</a> <span class="p1">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="ami_3.html#K16">il.</a> <font color="Maroon">j</font></span>)</span></span>)</span>,<font color="Maroon">k</font>
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E21:20_1_1"/>
<font color="Maroon">IT</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">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="reloc.html#K3" target="_self">IncAddr</a> <span class="p1">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="ami_3.html#K16">il.</a> <font color="Maroon">m</font></span>)</span></span>)</span>,<font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="reloc.html#E1:20_1_1"><i><font color="Green">E24</font></i></a>, </i>;<br/>


</div>
