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

<b>let </b><font color="Maroon">s</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/><b>let </b><font color="Maroon">a</font> be  <a href="sf_mastr.html#V1">read-write</a>  <a href="scmfsa_2.html#M1">Int-Location</a> ;<br/><b>let </b><font color="Maroon">bb</font> be    <a href="scmfsa_2.html#M1">Int-Location</a> ;<br/><b>let </b><font color="Maroon">cc</font> be    <a href="scmfsa_2.html#M1">Int-Location</a> ;<br/><b>let </b><font color="Maroon">Ig</font> be  <a href="scmfsa7b.html#V1">good</a> <a href="scmfsa6a.html#NM1">Macro-Instruction</a>;<br/>









<b>set </b><font color="Maroon">D</font> = <a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="subset_1.html#K4">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> ;<br/>
<b>set </b><font color="Maroon">I</font> = <font color="Maroon">Ig</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:33"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">s</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="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E2:33"/><i><font color="Green">E35</font></i>: 
 <a href="sfmastr3.html#R3" target="_self">ProperForUpBody</a> <font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font>,<font color="Maroon">Ig</font>,<font color="Maroon">s</font>
 ;<br/>

<b>set </b><font color="Maroon">SF</font> =  <a href="sfmastr3.html#K2" target="_self">StepForUp</a> <font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font>,<font color="Maroon">Ig</font>,<font color="Maroon">s</font>;<br/>
<b>set </b><font color="Maroon">aux</font> = 1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p1">(<span class="default"><span class="p2"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">s2</font> = <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa6b.html#K2">+*</a> <span class="p2">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p3">(<span class="default"><span class="p4"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">cc</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="scmfsa6b.html#K2">+*</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">bb</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">IB</font> = <span class="p1">(<span class="default"><font color="Maroon">Ig</font> <a href="scmfsa6a.html#K6">';'</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K9">AddTo</a> <font color="Maroon">a</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K6">';'</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K10">SubFrom</a> <span class="p2">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p3">(<span class="default"><span class="p4"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">SW2</font> =  <a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ig</font> <a href="scmfsa6a.html#K6">';'</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K9">AddTo</a> <font color="Maroon">a</font>,<span class="p4">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K6">';'</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K10">SubFrom</a> <span class="p3">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p4">(<span class="default"><span class="p5"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p5">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa6b.html#K2">+*</a> <span class="p3">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p4">(<span class="default"><span class="p5"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p5">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">cc</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="scmfsa6b.html#K2">+*</a> <font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">bb</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">scb1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">cc</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K2" target="_self">StepForUp</a> <font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font>,<font color="Maroon">Ig</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">cc</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 );<br/>
<a NAME="E3:33"/><i><font color="Green">E36</font></i>: 
1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p1">(<span class="default"><span class="p2"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="scmfsa_2.html#T66">SCMFSA_2:66</a></i>;<br/>
<a NAME="E4:33"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E5:33"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:33"/><b>then </b><i><font color="Green">E37</font></i>: 
1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p1">(<span class="default"><span class="p2"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="sfmastr1.html#T21">SFMASTR1:21</a></i>;<br/>
<a NAME="E7:33"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K2" target="_self">StepForUp</a> <font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font>,<font color="Maroon">Ig</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 0</span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:33_1"/>
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <span class="p2">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p3">(<span class="default"><span class="p4"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Ig</font> <a href="scmfsa6a.html#K6">';'</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K9">AddTo</a> <font color="Maroon">a</font>,<span class="p5">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K6">';'</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K10">SubFrom</a> <span class="p4">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p5">(<span class="default"><span class="p0"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p0">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa6b.html#K2">+*</a> <span class="p4">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p5">(<span class="default"><span class="p0"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p0">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">cc</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="scmfsa6b.html#K2">+*</a> <font color="Maroon">a</font>,<span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">bb</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa6b.html#K2">+*</a> <span class="p2">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p3">(<span class="default"><span class="p4"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">cc</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="scmfsa6b.html#K2">+*</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">bb</font></span>)</span>
 
<b>by </b><i><a class="ref" href="scmfsa_9.html#D5">SCMFSA_9:def 5</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K2" target="_self">StepForUp</a> <font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font>,<font color="Maroon">Ig</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 0</span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa6b.html#K2">+*</a> <span class="p2">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p3">(<span class="default"><span class="p4"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">cc</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sfmastr3.html#E1:33"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">cc</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">bb</font></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/>
<b>hence </b><a NAME="E4:33_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">cc</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="txt" href="sfmastr3.html#E3:33"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:33"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 <b>and </b><br/><a NAME="E2:33_2"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K2" target="_self">StepForUp</a> <font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font>,<font color="Maroon">Ig</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E3:33_2"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K2" target="_self">StepForUp</a> <font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font>,<font color="Maroon">Ig</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_2_1"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K2" target="_self">StepForUp</a> <font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font>,<font color="Maroon">Ig</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E2:33_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K2" target="_self">StepForUp</a> <font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font>,<font color="Maroon">Ig</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="subset_1.html#K4">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K2" target="_self">StepForUp</a> <font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font>,<font color="Maroon">Ig</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="subset_1.html#K4">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="scmfsa9a.html#T37">SCMFSA9A:37</a></i>;<br/>
<b>hence </b><a NAME="E3:33_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sfmastr3.html#E7:33"><i><font color="Green">E38</font></i></a>, <a class="txt" href="sfmastr3.html#E8:33"><i><font color="Green">E40</font></i></a>, <a class="ref" href="scmfsa6a.html#T38">SCMFSA6A:38</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:33_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">cc</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="txt" href="sfmastr3.html#E6:33"><i><font color="Green">E37</font></i></a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">scb1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">cc</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E6:33_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">scb1</font>
 
<b>by </b><i><a class="txt" href="sfmastr3.html#E6:33"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>assume </b><a NAME="E7:33_2"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">cc</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<a NAME="E8:33_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sfmastr3.html#K2" target="_self">StepForUp</a> <font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font>,<font color="Maroon">Ig</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p2">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p3">(<span class="default"><span class="p4"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> <font color="Maroon">scb1</font>
 
<b>by </b><i><a class="txt" href="sfmastr3.html#E7:33"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E9:33_2"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="sfmastr3.html#E8:33"><i><font color="Green">E40</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:33"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="sfmastr3.html#E2:33"><i><font color="Green">E35</font></i></a>, <a class="txt" href="sfmastr3.html#E3:33"><i><font color="Green">E36</font></i></a>);<br/></i>
<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>thus </b><a NAME="E10:33"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 <b>by </b><i><a class="txt" href="sfmastr3.html#E6:33"><i><font color="Green">E37</font></i></a></i>;<br/>

<b>assume </b><a NAME="E11:33"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">cc</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<a NAME="E12:33"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sfmastr3.html#K2" target="_self">StepForUp</a> <font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font>,<font color="Maroon">Ig</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p2">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p3">(<span class="default"><span class="p4"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">cc</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E13:33"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">cc</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="sfmastr3.html#E7:33"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>hence </b><a NAME="E14:33"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K2" target="_self">StepForUp</a> <font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font>,<font color="Maroon">Ig</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">bb</font>,<font color="Maroon">cc</font></span><a href="sfmastr1.html#K3">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">Ig</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/>


</div>
