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

<b>set </b><font color="Maroon" title="c2">D</font> = <a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> ;<br/>
<b>let </b><font color="Maroon" title="c3">s</font> be   <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ;<br/><b>let </b><font color="Maroon" title="c4">I</font> be   <a href="scmfsa6a.html#NM1" title="SCMFSA6A:NM.1">Macro-Instruction</a>;<br/><b>let </b><font color="Maroon" title="c5">a</font> be  <a href="sf_mastr.html#NV2" title="SF_MASTR:NV.2">read-write</a>  <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> ;<br/>





<b>assume </b><a NAME="E1:48"/><i><font color="Green" title="E41">A1</font></i>: 
<font color="Maroon" title="c4">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c3">s</font>
 ;<br/>

<b>assume </b><a NAME="E2:48"/><i><font color="Green" title="E42">A2</font></i>: 
<font color="Maroon" title="c4">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c3">s</font>
 ;<br/>

<b>assume </b><a NAME="E3:48"/><i><font color="Green" title="E43">A3</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<b>set </b><font color="Maroon" title="c6">s1</font> = <font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">s2</font> = <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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 1;<br/>
<b>set </b><font color="Maroon" title="c8">C1</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">i</font> = <font color="Maroon" title="c5">a</font> <a href="scmfsa_2.html#K15" title="SCMFSA_2:func.15">&gt;0_goto</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 4</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">sI</font> = <font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E4:48"/><i><font color="Green" title="E44">A4</font></i>: 
 <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span>
 
<b>by </b><i><a class="ref" href="scmfsa_9.html#T10" target="_self" title="SCMFSA_9:th.10">Th10</a></i>;<br/>
<a NAME="E5:48"/>
 <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="scmfsa8a.html#T9" title="SCMFSA8A:th.9">SCMFSA8A:9</a></i>;<br/>
<a NAME="E6:48"/><b>then </b><i><font color="Green" title="E45">A5</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/>
<a NAME="E7:48"/><i><font color="Green" title="E46">A6</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#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sf_mastr.html#T65" title="SF_MASTR:th.65">SF_MASTR:65</a></i>;<br/>
<i><font color="Green" title="E47">A7</font></i>:  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><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> </span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D15" title="AMI_1:def.15">AMI_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><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> </span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_9.html#E7:48"><i><font color="Green" title="E46">A6</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0
<b>by </b><i><a class="ref" href="sf_mastr.html#T66" title="SF_MASTR:th.66">SF_MASTR:66</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></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="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_9.html#E4:48"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="scmfsa_9.html#E6:48"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_9.html#E4:48"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="scmfsa6b.html#T7" title="SCMFSA6B:th.7">SCMFSA6B:7</a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="scmfsa_2.html#K15" title="SCMFSA_2:func.15">&gt;0_goto</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 4</span>)</span>
<b>by </b><i><a class="ref" href="scmfsa_9.html#T11" target="_self" title="SCMFSA_9:th.11">Th11</a></i>
;<br/>
<a NAME="E10:48"/><b>then </b><i><font color="Green" title="E48">A8</font></i>: 
 <a href="ami_1.html#K8" title="AMI_1:func.8">CurInstr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="scmfsa_2.html#K15" title="SCMFSA_2:func.15">&gt;0_goto</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 4</span>)</span>
 
<b>by </b><i><a class="txt" href="scmfsa_9.html#E8:48"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="ami_1.html#D17" title="AMI_1:def.17">AMI_1:def 17</a></i>;<br/>
<i><font color="Green" title="E49">A9</font></i>: <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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D19" title="AMI_1:def.19">AMI_1:def 19</a></i>
<br/>.= 
 <a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D19" title="AMI_1:def.19">AMI_1:def 19</a></i>
<br/>.= 
 <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="scmfsa_2.html#K15" title="SCMFSA_2:func.15">&gt;0_goto</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 4</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_9.html#E10:48"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="ami_1.html#D18" title="AMI_1:def.18">AMI_1:def 18</a></i>
;<br/>
<a NAME="E12:48"/>
( not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">s</font> )
 
<b>by </b><i><a class="ref" href="scmfsa6b.html#T12" title="SCMFSA6B:th.12">SCMFSA6B:12</a>, <a class="ref" href="scmfsa_2.html#T66" title="SCMFSA_2:th.66">SCMFSA_2:66</a></i>;<br/>
<a NAME="E13:48"/><b>then </b><i><font color="Green" title="E50">A10</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/>
<i><font color="Green" title="E51">A11</font></i>:  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><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> </span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D15" title="AMI_1:def.15">AMI_1:def 15</a></i>
<br/>.= 
 <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 4
<b>by </b><i><a class="txt" href="scmfsa_9.html#E3:48"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="scmfsa_9.html#E11:48"><i><font color="Green" title="E49">A9</font></i></a>, <a class="txt" href="scmfsa_9.html#E13:48"><i><font color="Green" title="E50">A10</font></i></a>, <a class="ref" href="scmfsa_2.html#T97" title="SCMFSA_2:th.97">SCMFSA_2:97</a></i>
;<br/>
<a NAME="E15:48"/>
( ( for <font color="Olive" title="b1">c</font> being   <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a>  holds  <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 1</span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Olive" title="b1">c</font> ) &amp; ( for <font color="Olive" title="b1">f</font> being   <a href="scmfsa_2.html#M2" title="SCMFSA_2:mode.2">FinSeq-Location</a>  holds  <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 1</span>)</span> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Olive" title="b1">f</font> ) )
 
<b>by </b><i><a class="txt" href="scmfsa_9.html#E11:48"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="scmfsa_2.html#T97" title="SCMFSA_2:th.97">SCMFSA_2:97</a></i>;<br/>
<b>then </b><i><font color="Green" title="E52">A12</font></i>: <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 1</span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> </span>)</span>
<b>by </b><i><a class="ref" href="scmfsa6a.html#T38" title="SCMFSA6A:th.38">SCMFSA6A:38</a></i>
<br/>.= 
<font color="Maroon" title="c3">s</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> </span>)</span>
<b>by </b><i><a class="ref" href="scmfsa8a.html#T11" title="SCMFSA8A:th.11">SCMFSA8A:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> </span>)</span>
<b>by </b><i><a class="ref" href="scmfsa8a.html#T11" title="SCMFSA8A:th.11">SCMFSA8A:11</a></i>
;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> implies (  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> $1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p0">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> $1</span>)</span></span>)</span> <a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">+</a> 4 &amp; <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> $1</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> $1</span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> </span>)</span> ) );<br/>
<a NAME="E17:48"/><i><font color="Green" title="E53">A13</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_6"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:48_6"/><i><font color="Green" title="E54">A14</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#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sf_mastr.html#T65" title="SF_MASTR:th.65">SF_MASTR:65</a></i>;<br/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 0</span>)</span> = 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D19" title="AMI_1:def.19">AMI_1:def 19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><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> </span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D15" title="AMI_1:def.15">AMI_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><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> </span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_9.html#E2:48_6"><i><font color="Green" title="E54">A14</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0
<b>by </b><i><a class="ref" href="sf_mastr.html#T66" title="SF_MASTR:th.66">SF_MASTR:66</a></i>
;<br/>
<a NAME="E4:48_6"/><b>then </b>
<span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p0">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 0</span>)</span></span>)</span> <a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">+</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span>
 
<b>by </b><i><a class="ref" href="scmfsa_4.html#D1" title="SCMFSA_4:def.1">SCMFSA_4:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:48_6"/>
(  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p0">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 0</span>)</span></span>)</span> <a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">+</a> 4 &amp; <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 0</span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="scmfsa_9.html#E14:48"><i><font color="Green" title="E51">A11</font></i></a>, <a class="txt" href="scmfsa_9.html#E16:48"><i><font color="Green" title="E52">A12</font></i></a>, <a class="ref" href="ami_1.html#D19" title="AMI_1:def.19">AMI_1:def 19</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E54">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:48_7"/><i><font color="Green" title="E55">A16</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">k</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_7_1"/><i><font color="Green" title="E56">A17</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:48_7_1"/>
<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:48_7_1"/><b>then </b>
<font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa_9.html#E1:48_7_1"><i><font color="Green" title="E56">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E4:48_7_1"/>
(  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p0">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">+</a> 4 &amp; <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="scmfsa_9.html#E1:48"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="scmfsa_9.html#E2:48"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="scmfsa_9.html#E1:48_7"><i><font color="Green" title="E55">A16</font></i></a>, <a class="ref" href="scmfsa_9.html#T44" target="_self" title="SCMFSA_9:th.44">Th44</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:48_7"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c11">s2</font> = <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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c12">loc4</font> =  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c13">s3</font> = <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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p0">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<a NAME="E19:48"/><i><font color="Green" title="E55">A18</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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">k</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="scmfsa_9.html#E17:48"><i><font color="Green" title="E53">A13</font></i></a>, <a class="txt" href="scmfsa_9.html#E18:48"><i><font color="Green" title="E54">A15</font></i></a>);<br/></i>
<a NAME="E20:48"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[ <a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>]
 
;<br/>
<a NAME="E21:48"/><b>then </b><i><font color="Green" title="E56">A19</font></i>: 
 <a href="ami_1.html#K8" title="AMI_1:func.8">CurInstr</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p0">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K13" title="SCMFSA_2:func.13">goto</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="scmfsa_9.html#E1:48"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="scmfsa_9.html#E2:48"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="scmfsa_9.html#T45" target="_self" title="SCMFSA_9:th.45">Th45</a></i>;<br/>
<i><font color="Green" title="E57">A20</font></i>: <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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p0">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p0">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D19" title="AMI_1:def.19">AMI_1:def 19</a></i>
<br/>.= 
 <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K13" title="SCMFSA_2:func.13">goto</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p0">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_9.html#E21:48"><i><font color="Green" title="E56">A19</font></i></a>, <a class="ref" href="ami_1.html#D18" title="AMI_1:def.18">AMI_1:def 18</a></i>
;<br/>
<a NAME="E23:48"/><i><font color="Green" title="E58">A21</font></i>: 
 <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span>
 
<b>by </b><i><a class="ref" href="scmfsa_9.html#T38" target="_self" title="SCMFSA_9:th.38">Th38</a></i>;<br/>
<i><font color="Green" title="E59">A22</font></i>:  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><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> </span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D15" title="AMI_1:def.15">AMI_1:def 15</a></i>
<br/>.= 
 <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_9.html#E22:48"><i><font color="Green" title="E57">A20</font></i></a>, <a class="ref" href="scmfsa_2.html#T95" title="SCMFSA_2:th.95">SCMFSA_2:95</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c14">s4</font> = <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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></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="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_9.html#E6:48"><i><font color="Green" title="E45">A5</font></i></a>, <a class="txt" href="scmfsa_9.html#E23:48"><i><font color="Green" title="E58">A21</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_9.html#E23:48"><i><font color="Green" title="E58">A21</font></i></a>, <a class="ref" href="scmfsa6b.html#T7" title="SCMFSA6B:th.7">SCMFSA6B:7</a></i>
<br/>.= 
 <a href="scmfsa_2.html#K13" title="SCMFSA_2:func.13">goto</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="scmfsa_9.html#T46" target="_self" title="SCMFSA_9:th.46">Th46</a></i>
;<br/>
<a NAME="E26:48"/><b>then </b><i><font color="Green" title="E60">A23</font></i>: 
 <a href="ami_1.html#K8" title="AMI_1:func.8">CurInstr</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K13" title="SCMFSA_2:func.13">goto</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="scmfsa_9.html#E24:48"><i><font color="Green" title="E59">A22</font></i></a>, <a class="ref" href="ami_1.html#D17" title="AMI_1:def.17">AMI_1:def 17</a></i>;<br/>
<i><font color="Green" title="E61">A24</font></i>: <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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D19" title="AMI_1:def.19">AMI_1:def 19</a></i>
<br/>.= 
 <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K13" title="SCMFSA_2:func.13">goto</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span>,<span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_9.html#E26:48"><i><font color="Green" title="E60">A23</font></i></a>, <a class="ref" href="ami_1.html#D18" title="AMI_1:def.18">AMI_1:def 18</a></i>
;<br/>
<i><font color="Green" title="E62">A25</font></i>:  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p5">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p5">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><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> </span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D15" title="AMI_1:def.15">AMI_1:def 15</a></i>
<br/>.= 
 <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0
<b>by </b><i><a class="txt" href="scmfsa_9.html#E27:48"><i><font color="Green" title="E61">A24</font></i></a>, <a class="ref" href="scmfsa_2.html#T95" title="SCMFSA_2:th.95">SCMFSA_2:95</a></i>
;<br/>
<a NAME="E29:48"/><i><font color="Green" title="E63">A26</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p0">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
;<br/>
<b>thus </b><a NAME="E30:48"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p0">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0
 <b>by </b><i><a class="txt" href="scmfsa_9.html#E28:48"><i><font color="Green" title="E62">A25</font></i></a></i>;<br/>

<div><i><font color="Green" title="E64">A27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_13"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:48_13"/><i><font color="Green" title="E65">A28</font></i>: 
<font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3
 ;<br/><b>assume </b><a NAME="E2:48_13"/>
<font color="Maroon" title="c15">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon" title="c16">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:48_13"/><i><font color="Green" title="E66">A29</font></i>: 
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">n</font> = <font color="Maroon" title="c16">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E6:48_13"/>
( <font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E7:48_13"/><b>then </b><i><font color="Green" title="E67">A30</font></i>: 
( <font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_13_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_13_1"/>
( <font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3 )
 </span><b>by </b><i><a class="txt" href="scmfsa_9.html#E29:48"><i><font color="Green" title="E63">A26</font></i></a>, <a class="txt" href="scmfsa_9.html#E7:48_13"><i><font color="Green" title="E67">A30</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_13_1_1"><b>suppose </b></a><a NAME="E1:48_13_1_1"/>
<font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:48_13_1_1"/><b>then </b>
<font color="Maroon" title="c17">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa_9.html#E4:48_13"><i><font color="Green" title="E66">A29</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:48_13_1_1"/><b>then </b><i><font color="Green" title="E68">A31</font></i>: 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p0">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c17">n</font></span>)</span></span>)</span> <a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">+</a> 4
 <b>by </b><i><a class="txt" href="scmfsa_9.html#E19:48"><i><font color="Green" title="E55">A18</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c18">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:48_13_1_1"/><i><font color="Green" title="E69">A32</font></i>: 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c18">m</font>
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T21" title="SCMFSA_2:th.21">SCMFSA_2:21</a></i>;<br/><a NAME="E6:48_13_1_1"/>
 <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c18">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">I</font>
 <b>by </b><i><a class="txt" href="scmfsa_9.html#E1:48"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="scmfsa_9.html#E5:48_13_1_1"><i><font color="Green" title="E69">A32</font></i></a>, <a class="ref" href="scmfsa7b.html#D7" title="SCMFSA7B:def.7">SCMFSA7B:def 7</a></i>;<br/><a NAME="E7:48_13_1_1"/><b>then </b>
<font color="Maroon" title="c18">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">I</font>
 <b>by </b><i><a class="ref" href="scmfsa6a.html#T15" title="SCMFSA6A:th.15">SCMFSA6A:15</a></i>;<br/><a NAME="E8:48_13_1_1"/><b>then </b><i><font color="Green" title="E70">A33</font></i>: 
<font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 6
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E9:48_13_1_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 6
 <b>by </b><i><a class="ref" href="scmfsa_9.html#T5" target="_self" title="SCMFSA_9:th.5">Th5</a></i>;<br/><a NAME="E10:48_13_1_1"/><b>then </b>
 <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa_9.html#E8:48_13_1_1"><i><font color="Green" title="E70">A33</font></i></a>, <a class="ref" href="scmfsa6a.html#T15" title="SCMFSA6A:th.15">SCMFSA6A:15</a></i>;<br/><b>hence </b><a NAME="E11:48_13_1_1"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa_9.html#E4:48_13"><i><font color="Green" title="E66">A29</font></i></a>, <a class="txt" href="scmfsa_9.html#E3:48_13_1_1"><i><font color="Green" title="E68">A31</font></i></a>, <a class="txt" href="scmfsa_9.html#E5:48_13_1_1"><i><font color="Green" title="E69">A32</font></i></a>, <a class="ref" href="scmfsa_4.html#D1" title="SCMFSA_4:def.1">SCMFSA_4:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_13_1_2"><b>suppose </b></a><a NAME="E1:48_13_1_2"/>
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_13_1_2"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa_9.html#E24:48"><i><font color="Green" title="E59">A22</font></i></a>, <a class="ref" href="scmfsa_9.html#T38" target="_self" title="SCMFSA_9:th.38">Th38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_13_1_3"><b>suppose </b></a><a NAME="E1:48_13_1_3"/>
<font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3
 ;<br/><div class="add"><a NAME="E2:48_13_1_3"/><b>then </b>
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3
 <b>by </b><i><a class="txt" href="scmfsa_9.html#E1:48_13"><i><font color="Green" title="E65">A28</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E3:48_13_1_3"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa_9.html#E28:48"><i><font color="Green" title="E62">A25</font></i></a>, <a class="ref" href="scmfsa_9.html#T10" target="_self" title="SCMFSA_9:th.10">Th10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_14"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:48_14"/><i><font color="Green" title="E65">A34</font></i>: 
<font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_14_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_14_1"/>
( <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c15">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_14_1_1"><b>suppose </b></a><a NAME="E1:48_14_1_1"/>
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_14_1_1"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa_9.html#E4:48"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="scmfsa_9.html#E8:48"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="ami_1.html#D19" title="AMI_1:def.19">AMI_1:def 19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_14_1_2"><b>suppose </b></a><a NAME="E1:48_14_1_2"/>
<font color="Maroon" title="c15">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_14_1_2"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa_9.html#E31:48"><i><font color="Green" title="E64">A27</font></i></a>, <a class="txt" href="scmfsa_9.html#E1:48_14"><i><font color="Green" title="E65">A34</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E33:48"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3 holds <br/> <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><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="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">I</font></span>)</span>
 ;<br/>


</div>
