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

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="scmfsa6a.html#NM1">Macro-Instruction</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/>



<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p3">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> ;<br/>
<b>assume </b><a NAME="E1:62"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="scmfsa7b.html#R5">is_closed_on</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:62"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="scmfsa7b.html#R6">is_halting_on</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p5">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E3:62"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p5">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span> is <a href="ami_1.html#V9">halting</a>
 
<b>by </b><i><a class="txt" href="scmfsa8a.html#E1:62"><i><font color="Green">E55</font></i></a>, <a class="txt" href="scmfsa8a.html#E2:62"><i><font color="Green">E56</font></i></a>, <a class="txt" href="scmfsa8a.html#E67"><i><font color="Green">Lemma54</font></i></a></i>;<br/>
<a NAME="E4:62"/><i><font color="Green">E58</font></i>: 
 <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p0">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="txt" href="scmfsa8a.html#E1:62"><i><font color="Green">E55</font></i></a>, <a class="txt" href="scmfsa8a.html#E2:62"><i><font color="Green">E56</font></i></a>, <a class="txt" href="scmfsa8a.html#E67"><i><font color="Green">Lemma54</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:62_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> or <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E2:62_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:62_1"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><i><font color="Green">E60</font></i>:  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p4">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p4">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="scmfsa8a.html#T17" target="_self">Th17</a>, <a class="ref" href="scmfsa6a.html#T61">SCMFSA6A:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p4">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="scmfsa6a.html#T61">SCMFSA6A:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p4">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="scmfsa6a.html#T61">SCMFSA6A:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="scmfsa8a.html#T29" target="_self">Th29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/><a NAME="E5:62_1"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E6:62_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E7:62_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p4">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa8a.html#E4:62_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_1_2"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 )
 </span><b>by </b><i><a class="txt" href="scmfsa8a.html#E3:62_1"><i><font color="Green">E59</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_2_1"><b>suppose </b></a><a NAME="E1:62_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:62_1_2_1"/><b>then </b>
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p2">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa8a.html#E1:62"><i><font color="Green">E55</font></i></a>, <a class="txt" href="scmfsa8a.html#E2:62"><i><font color="Green">E56</font></i></a>, <a class="txt" href="scmfsa8a.html#E67"><i><font color="Green">Lemma54</font></i></a></i>;<br/><a NAME="E3:62_1_2_1"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p2">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="scmfsa8a.html#E1:62"><i><font color="Green">E55</font></i></a>, <a class="ref" href="scmfsa7b.html#D7">SCMFSA7B:def 7</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:62_1_2_1"/><i><font color="Green">E63</font></i>: 
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p2">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T21">SCMFSA_2:21</a></i>;<br/><a NAME="E6:62_1_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="scmfsa8a.html#E3:62_1_2_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="scmfsa8a.html#E5:62_1_2_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="scmfsa6a.html#T15">SCMFSA6A:15</a></i>;<br/><a NAME="E7:62_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p4">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa8a.html#E7:62_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E8:62_1_2_1"/>
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p2">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p4">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa8a.html#E5:62_1_2_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="scmfsa6a.html#T15">SCMFSA6A:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_2_2"><b>suppose </b></a><a NAME="E1:62_1_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:62_1_2_2"/><b>then </b>
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p2">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa8a.html#E1:62"><i><font color="Green">E55</font></i></a>, <a class="txt" href="scmfsa8a.html#E2:62"><i><font color="Green">E56</font></i></a>, <a class="txt" href="scmfsa8a.html#E67"><i><font color="Green">Lemma54</font></i></a></i>;<br/><b>hence </b><a NAME="E3:62_1_2_2"/>
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p2">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p4">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa8a.html#E7:62_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="scmfsa6a.html#T15">SCMFSA6A:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_2_3"><b>suppose </b></a><a NAME="E1:62_1_2_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 ;<br/><div class="add"><a NAME="E2:62_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p0">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa8a.html#E1:62"><i><font color="Green">E55</font></i></a>, <a class="txt" href="scmfsa8a.html#E2:62"><i><font color="Green">E56</font></i></a>, <a class="txt" href="scmfsa8a.html#E67"><i><font color="Green">Lemma54</font></i></a></i>;<br/><b>then </b><i><font color="Green">E62</font></i>:  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p2">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p2">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p2">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa8a.html#E3:62"><i><font color="Green">E57</font></i></a>, <a class="ref" href="scmfsa8a.html#T5" target="_self">Th5</a></i>
<br/>.= 
 <a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="scmfsa8a.html#E1:62"><i><font color="Green">E55</font></i></a>, <a class="txt" href="scmfsa8a.html#E2:62"><i><font color="Green">E56</font></i></a>, <a class="txt" href="scmfsa8a.html#E4:62"><i><font color="Green">E58</font></i></a>, <a class="txt" href="scmfsa8a.html#E67"><i><font color="Green">Lemma54</font></i></a></i>
;<br/><a NAME="E4:62_1_2_3"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p4">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="scmfsa8a.html#E4:62_1"><i><font color="Green">E60</font></i></a></i>;<br/><a NAME="E5:62_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p4">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E6:62_1_2_3"/>
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p2">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p4">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a> </span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa8a.html#E3:62_1_2_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="scmfsa6a.html#T15">SCMFSA6A:15</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:62"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p3">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a>  <a href="scmfsa7b.html#R5">is_closed_on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="scmfsa7b.html#D7">SCMFSA7B:def 7</a></i>;<br/>

<b>thus </b><a NAME="E7:62"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmfsa6a.html#K4">';'</a> <span class="p3">(<span class="default"><a href="scmfsa8a.html#K2" target="_self">Goto</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa6a.html#K4">';'</a> <a href="scmfsa_4.html#K5">SCM+FSA-Stop</a>  <a href="scmfsa7b.html#R6">is_halting_on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="scmfsa8a.html#E3:62"><i><font color="Green">E57</font></i></a>, <a class="ref" href="scmfsa7b.html#D8">SCMFSA7B:def 8</a></i>;<br/>


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