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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="scmfsa6a.html#NM1">Macro-Instruction</a>;<br/>

<b>assume </b><a NAME="E1:27_2_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="scmfsa6b.html#V3" target="_self">keeping_0</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="scmfsa6b.html#D2" target="_self">SCMFSA6B:def 2</a><br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E2:27_2_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:27_2_1"/><i><font color="Green">E23</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 
<b>by </b><i><a class="ref" href="ami_3.html#D13">AMI_3:def 13</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means not  <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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>assume </b><a NAME="E4:27_2_1"/>
not  <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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E5:27_2_1"/><b>then </b><i><font color="Green">E24</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:27_2_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <b>and </b><br/><a NAME="E8:27_2_1"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="scmfsa6b.html#E5:27_2_1"><i><font color="Green">E24</font></i></a>);<br/></i>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="sf_mastr.html#K6">FirstNotUsed</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="scmfsa6b.html#K1" target="_self">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> <a href="scmfsa_2.html#K8">:=</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K6">FirstNotUsed</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> as   <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="scmfsa6b.html#K2" target="_self">+*</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K6">FirstNotUsed</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="scmfsa_2.html#K20">.</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> as   <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a>  ;<br/>
<a NAME="E11:27_2_1"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="scmfsa6b.html#V3" target="_self">keeping_0</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="scmfsa6b.html#D4" target="_self">SCMFSA6B:def 4</a><br/>

<b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span>;<br/>
<i><font color="Green">E27</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</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> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</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>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="ami_3.html#T34">AMI_3:34</a></i>
;<br/>
<a NAME="E2:27_2_1_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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 
<b>by </b><i><a class="ref" href="ami_1.html#T48">AMI_1:48</a></i>;<br/>
<a NAME="E3:27_2_1_1"/><b>then </b>
not  <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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:27_2_1_1"/><b>then </b>
not  <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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</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"><font color="Maroon">c<sub>1</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>
 
<b>by </b><i><a class="txt" href="scmfsa6b.html#E7:27_2_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="scmfsa6b.html#E1:27_2_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:27_2_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="scmfsa6b.html#E2:27_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="scmfsa6a.html#T1">SCMFSA6A:1</a></i>;<br/>
<a NAME="E6:27_2_1_1"/><i><font color="Green">E29</font></i>: 
not  <a href="sf_mastr.html#K6">FirstNotUsed</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="scmfsa6b.html#E3:27_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="scmfsa_2.html#T84">SCMFSA_2:84</a></i>;<br/>
<a NAME="E7:27_2_1_1"/>
 <a href="sf_mastr.html#K6">FirstNotUsed</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 
<b>by </b><i><a class="ref" href="scmfsa_2.html#T81">SCMFSA_2:81</a></i>;<br/>
<a NAME="E8:27_2_1_1"/><b>then </b>
not  <a href="sf_mastr.html#K6">FirstNotUsed</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:27_2_1_1"/><b>then </b>
not  <a href="sf_mastr.html#K6">FirstNotUsed</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</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>
 
<b>by </b><i><a class="txt" href="scmfsa6b.html#E1:27_2_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="scmfsa6b.html#E6:27_2_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E10:27_2_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="ami_1.html#K17">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="scmfsa6b.html#E5:27_2_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="scmfsa6a.html#T1">SCMFSA6A:1</a></i>;<br/>

<b>take </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1;<br/>

<b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E11:27_2_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> holds <br/> <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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="scmfsa6b.html#E8:27_2_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E12:27_2_1_1"/><i><font color="Green">E32</font></i>: 
not  <a href="sf_mastr.html#K6">FirstNotUsed</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="sf_mastr.html#T54">SF_MASTR:54</a></i>;<br/>
<a NAME="E13:27_2_1_1"/><i><font color="Green">E33</font></i>: 
not  <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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_2_1_1_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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:27_2_1_1_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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is    <a href="scmfsa_2.html#M1">Int-Location</a> 
 
<b>by </b><i><a class="ref" href="scmfsa_2.html#T11">SCMFSA_2:11</a></i>;<br/>
<b>hence </b><a NAME="E3:27_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T84">SCMFSA_2:84</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E34</font></i>: <font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<font color="Maroon">c<sub>7</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa6b.html#E12:27_2_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="scmfsa6a.html#T4">SCMFSA6A:4</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa6b.html#E13:27_2_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="scmfsa6a.html#T4">SCMFSA6A:4</a></i>
;<br/>
<a NAME="E15:27_2_1_1"/><i><font color="Green">E35</font></i>: 
not  <a href="sf_mastr.html#K6">FirstNotUsed</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="sf_mastr.html#K4">UsedInt*Loc</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_2_1_1_4"/>
 <a href="sf_mastr.html#K6">FirstNotUsed</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="sf_mastr.html#K4">UsedInt*Loc</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:27_2_1_1_4"/><b>then </b>
 <a href="sf_mastr.html#K6">FirstNotUsed</a> <font color="Maroon">c<sub>1</sub></font> is    <a href="scmfsa_2.html#M2">FinSeq-Location</a> 
 
<b>by </b><i><a class="ref" href="scmfsa_2.html#T12">SCMFSA_2:12</a></i>;<br/>
<b>hence </b><a NAME="E3:27_2_1_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T83">SCMFSA_2:83</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:27_2_1_1"/><i><font color="Green">E36</font></i>: 
not  <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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="sf_mastr.html#K4">UsedInt*Loc</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_2_1_1_5"/>
 <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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="sf_mastr.html#K4">UsedInt*Loc</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:27_2_1_1_5"/><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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is    <a href="scmfsa_2.html#M2">FinSeq-Location</a> 
 
<b>by </b><i><a class="ref" href="scmfsa_2.html#T12">SCMFSA_2:12</a></i>;<br/>
<b>hence </b><a NAME="E3:27_2_1_1_5"/>
contradiction
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T85">SCMFSA_2:85</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E37</font></i>: <font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<font color="Maroon">c<sub>7</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa6b.html#E15:27_2_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="scmfsa6a.html#T4">SCMFSA6A:4</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K4">UsedInt*Loc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa6b.html#E16:27_2_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="scmfsa6a.html#T4">SCMFSA6A:4</a></i>
;<br/>
<a NAME="E18:27_2_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> holds <br/> <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> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="scmfsa6b.html#E2:27_2_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="scmfsa6b.html#E10:27_2_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="scmfsa6b.html#E11:27_2_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="scmfsa6b.html#E14:27_2_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="sf_mastr.html#T73">SF_MASTR:73</a></i>;<br/>
<a NAME="E19:27_2_1_1"/><i><font color="Green">E39</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> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="scmfsa6b.html#E2:27_2_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="scmfsa6b.html#E10:27_2_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="scmfsa6b.html#E11:27_2_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="scmfsa6b.html#E14:27_2_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="scmfsa6b.html#E17:27_2_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="sf_mastr.html#T73">SF_MASTR:73</a></i>;<br/>
<a NAME="E20:27_2_1_1"/>
 <a href="sf_mastr.html#K6">FirstNotUsed</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="scmfsa_2.html#T66">SCMFSA_2:66</a></i>;<br/>
<a NAME="E21:27_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K6">FirstNotUsed</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="scmfsa_2.html#K20">.</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<a NAME="E22:27_2_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K6">FirstNotUsed</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="scmfsa_2.html#K20">.</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="txt" href="scmfsa6b.html#E10:27_2_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="scmfsa6b.html#E12:27_2_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="scmfsa6b.html#E18:27_2_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="sf_mastr.html#T69">SF_MASTR:69</a></i>;<br/>
<a NAME="E23:27_2_1_1"/><i><font color="Green">E41</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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="scmfsa_2.html#T5">SCMFSA_2:5</a></i>;<br/>
<a NAME="E24:27_2_1_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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="sf_mastr.html#K6">FirstNotUsed</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="scmfsa_2.html#T5">SCMFSA_2:5</a>, <a class="ref" href="scmfsa_2.html#T84">SCMFSA_2:84</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>8</sub></font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>7</sub></font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> <a href="scmfsa_2.html#K8">:=</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K6">FirstNotUsed</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa6b.html#E23:27_2_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>
;<br/>
<a NAME="E26:27_2_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> <a href="scmfsa_2.html#K8">:=</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K6">FirstNotUsed</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="scmfsa6b.html#E19:27_2_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ami_1.html#T54">AMI_1:54</a></i>;<br/>
<a NAME="E27:27_2_1_1"/><i><font color="Green">E43</font></i>: 
(  <a href="scmfsa_2.html#K4">intloc</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="scmfsa_2.html#K4">intloc</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="scmfsa_2.html#T66">SCMFSA_2:66</a>, <a class="ref" href="scmfsa_2.html#T84">SCMFSA_2:84</a></i>;<br/>
<i><font color="Green">E44</font></i>: <font color="Maroon">c<sub>8</sub></font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> = 
<font color="Maroon">c<sub>7</sub></font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="scmfsa6b.html#E27:27_2_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
;<br/>
<a NAME="E29:27_2_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="scmfsa_2.html#K20">.</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>10</sub></font> = 
 <a href="ami_1.html#K9">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D19">AMI_1:def 19</a></i>
<br/>.= 
 <a href="ami_1.html#K4">Exec</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8">CurInstr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

<br/>.= 
 <a href="ami_1.html#K4">Exec</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> <a href="scmfsa_2.html#K8">:=</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K6">FirstNotUsed</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa6b.html#E26:27_2_1_1"><i><font color="Green">E42</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E31:27_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa6b.html#E22:27_2_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="scmfsa6b.html#E28:27_2_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="scmfsa6b.html#E29:27_2_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="scmfsa_2.html#T89">SCMFSA_2:89</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:27_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="scmfsa6b.html#E1:27_2_1"><i><font color="Green">E21</font></i></a></i>;<br/>


</div>
