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

<b>let </b><font color="Maroon" title="c1">s2</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c1">s2</font> <a href="ami_1.html#K39" title="AMI_1:func.39">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span>,<span class="p1">(<span class="default"><a href="scmfsa_2.html#K12" title="SCMFSA_2:func.12">goto</a> <span class="p2">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span></span>)</span> is <a href="ami_1.html#V8" title="AMI_1:attr.8">halting</a></span><br/>

<b>set </b><font color="Maroon" title="c2">s1</font> = <font color="Maroon" title="c1">s2</font> <a href="ami_1.html#K39" title="AMI_1:func.39">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span>,<span class="p1">(<span class="default"><a href="scmfsa_2.html#K12" title="SCMFSA_2:func.12">goto</a> <span class="p2">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:19"/><i><font color="Green" title="E18">A1</font></i>: 
 <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font>
 
<b>by </b><i><a class="ref" href="ami_1.html#T48" title="AMI_1:th.48">AMI_1:48</a></i>;<br/>
<a NAME="E2:19"/><i><font color="Green" title="E19">A2</font></i>: 
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">s2</font>
 
<b>by </b><i><a class="ref" href="ami_1.html#T115" title="AMI_1:th.115">AMI_1:115</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b> <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="ami_1.html#K39" title="AMI_1:func.39">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K12" title="SCMFSA_2:func.12">goto</a> <span class="p4">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span></span>)</span></span>)</span>,$1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="ami_1.html#K39" title="AMI_1:func.39">+*</a> <span class="p2">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K12" title="SCMFSA_2:func.12">goto</a> <span class="p3">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E3:19"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="ami_1.html#T13" title="AMI_1:th.13">AMI_1:13</a></i>;<br/>
<a NAME="E4:19"/><i><font color="Green" title="E21">A4</font></i>: 
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="ami_1.html#K39" title="AMI_1:func.39">+*</a> <span class="p2">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K12" title="SCMFSA_2:func.12">goto</a> <span class="p3">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font>
 
<b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/>
<div><i><font color="Green" title="E22">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><b>then </b><i><font color="Green" title="E23">A6</font></i>:  <a href="ami_1.html#K8" title="AMI_1:func.8">CurInstr</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="ami_1.html#K39" title="AMI_1:func.39">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K12" title="SCMFSA_2:func.12">goto</a> <span class="p4">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="ami_1.html#K39" title="AMI_1:func.39">+*</a> <span class="p2">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K12" title="SCMFSA_2:func.12">goto</a> <span class="p3">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="ami_1.html#K39" title="AMI_1:func.39">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K12" title="SCMFSA_2:func.12">goto</a> <span class="p4">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T54" title="AMI_1:th.54">AMI_1:54</a></i>
<br/>.= 
 <a href="scmfsa_2.html#K12" title="SCMFSA_2:func.12">goto</a> <span class="p1">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:19"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="#E4:19"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>
;<br/> <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="ami_1.html#K39" title="AMI_1:func.39">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K12" title="SCMFSA_2:func.12">goto</a> <span class="p4">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="ami_1.html#K39" title="AMI_1:func.39">+*</a> <span class="p4">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span>,<span class="p4">(<span class="default"><a href="scmfsa_2.html#K12" title="SCMFSA_2:func.12">goto</a> <span class="p5">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T14" title="AMI_1:th.14">AMI_1:14</a></i>
<br/>.= 
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="ami_1.html#K39" title="AMI_1:func.39">+*</a> <span class="p2">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K12" title="SCMFSA_2:func.12">goto</a> <span class="p3">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:19"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="#E2:19_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="scmfsa_2.html#T95" title="SCMFSA_2:th.95">SCMFSA_2:95</a></i>
;<br/><b>hence </b><a NAME="E4:19_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E6:19"/><i><font color="Green" title="E23">A7</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E3:19"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="#E5:19"><i><font color="Green" title="E22">A5</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ami_1.html#D20" title="AMI_1:def.20">AMI_1:def 20</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not  <a href="ami_1.html#K8" title="AMI_1:func.8">CurInstr</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="ami_1.html#K39" title="AMI_1:func.39">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K12" title="SCMFSA_2:func.12">goto</a> <span class="p4">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span><br/>

 <a href="ami_1.html#K8" title="AMI_1:func.8">CurInstr</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="ami_1.html#K39" title="AMI_1:func.39">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K12" title="SCMFSA_2:func.12">goto</a> <span class="p4">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="ami_1.html#K39" title="AMI_1:func.39">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K12" title="SCMFSA_2:func.12">goto</a> <span class="p4">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="ami_1.html#K39" title="AMI_1:func.39">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K12" title="SCMFSA_2:func.12">goto</a> <span class="p4">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E6:19"><i><font color="Green" title="E23">A7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="ami_1.html#K39" title="AMI_1:func.39">+*</a> <span class="p2">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K12" title="SCMFSA_2:func.12">goto</a> <span class="p3">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="ami_1.html#K39" title="AMI_1:func.39">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K12" title="SCMFSA_2:func.12">goto</a> <span class="p4">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T54" title="AMI_1:th.54">AMI_1:54</a></i>
<br/>.= 
 <a href="scmfsa_2.html#K12" title="SCMFSA_2:func.12">goto</a> <span class="p1">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:19"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="#E4:19"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>
;<br/>
<b>hence </b><a NAME="E8:19"/>
 <a href="ami_1.html#K8" title="AMI_1:func.8">CurInstr</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="ami_1.html#K39" title="AMI_1:func.39">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K12" title="SCMFSA_2:func.12">goto</a> <span class="p4">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s2</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T47" title="SCMFSA_2:th.47">SCMFSA_2:47</a>, <a class="ref" href="scmfsa_2.html#T124" title="SCMFSA_2:th.124">SCMFSA_2:124</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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