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

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span>,<span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 1</span>)</span> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="ami_1.html#K5">halt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>;<br/>
<a NAME="E1:13_1_1"/><i><font color="Green">E5</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 1</span>)</span> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="ami_1.html#K5">halt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 1</span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T65">FUNCT_4:65</a></i>;<br/>
<a NAME="E2:13_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 1</span>)</span> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="ami_1.html#K5">halt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="funct_4.html#T65">FUNCT_4:65</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span>,<span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 1</span>)</span> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="ami_1.html#K5">halt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span> as  <a href="finset_1.html#V1">finite</a> <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="finset_1.html#T29">FINSET_1:29</a></i>;<br/>
<a NAME="E4:13_1_1"/><i><font color="Green">E6</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:13_1_1"/><i><font color="Green">E7</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 
<b>by </b><i><a class="txt" href="scmfsa6a.html#E1:13_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:13_1_1_1"/><i><font color="Green">E8</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></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="txt" href="scmfsa6a.html#E1:13_1_1"><i><font color="Green">E5</font></i></a></i>;<br/><b>assume </b><a NAME="E2:13_1_1_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="ami_1.html#NM2">Instruction-Location</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a>  <b>by </b><i><a class="txt" href="scmfsa6a.html#E1:13_1_1_1"><i><font color="Green">E8</font></i></a></i>;<br/><i><font color="Green">E10</font></i>: the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="ami_1.html#K3">ObjectKind</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="ami_1.html#D6">AMI_1:def 6</a></i>
<br/>.= 
the <a href="ami_1.html#U4">Instructions</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> 
<b>by </b><i><a class="ref" href="ami_1.html#D14">AMI_1:def 14</a></i>
;<br/><a NAME="E5:13_1_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E2:13_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:13_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="ami_1.html#K5">halt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="funct_4.html#T65">FUNCT_4:65</a></i>;<br/><a NAME="E7:13_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="ami_1.html#U4">Instructions</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E8:13_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E4:13_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="scmfsa6a.html#E5:13_1_1_1"><i><font color="Green">E11</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:13_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_1.html#K7">sproduct</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 
<b>by </b><i><a class="txt" href="scmfsa6a.html#E4:13_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="ami_1.html#D16">AMI_1:def 16</a>, <a class="txt" href="scmfsa6a.html#E5:13_1_1"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmfsa_2.html#K1">SCM+FSA</a>  <b>by </b><i><a class="ref" href="ami_1.html#D24">AMI_1:def 24</a></i>;<br/>
<a NAME="E9:13_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="scmfsa_4.html#V1">initial</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="ami_3.html#V1">programmed</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:13_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="scmfsa_4.html#V1">initial</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="scmfsa_4.html#D4">SCMFSA_4:def 4</a><br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:13_1_1_2_1"/><i><font color="Green">E8</font></i>: 
 <a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:13_1_1_2_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E3:13_1_1_2_1"/>
(  <a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K5">insloc</a> 0 or  <a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K5">insloc</a> 1 )
 
<b>by </b><i><a class="txt" href="scmfsa6a.html#E1:13_1_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="scmfsa6a.html#E1:13_1_1_2_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E4:13_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="scmfsa_2.html#T18">SCMFSA_2:18</a></i>;<br/>
<a NAME="E5:13_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="scmfsa6a.html#E2:13_1_1_2_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E6:13_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="txt" href="scmfsa6a.html#E2:13_1_1_2_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:13_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<b>hence </b><a NAME="E8:13_1_1_2_1"/>
 <a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E1:13_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E2:13_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></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="txt" href="scmfsa6a.html#E1:13_1_1"><i><font color="Green">E5</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ami_3.html#D13">AMI_3:def 13</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:13_1_1"/>
<span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span>,<span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 1</span>)</span> <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="ami_1.html#K5">halt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span> is   <a href="scmfsa6a.html#NM1">Macro-Instruction</a>
 ;<br/>


</div>
