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

<b>let </b><font color="Maroon" title="c1">s</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being <a href="sf_mastr.html#NV2" title="SF_MASTR:NV.2">read-write</a>  <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> <br/> for <font color="Olive" title="b2">bb</font>, <font color="Olive" title="b3">cc</font> being   <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> <br/> for <font color="Olive" title="b4">Ig</font> being <a href="scmfsa7b.html#V1" title="SCMFSA7B:attr.1">good</a> <a href="scmnorm.html#NM2" title="SCMNORM:NM.2">Program</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>   st <font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="sfmastr3.html#R1" title="SFMASTR3:pred.1">ProperForUpBody</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">bb</font>,<font color="Olive" title="b3">cc</font>,<font color="Olive" title="b4">Ig</font>,<font color="Maroon" title="c1">s</font> holds <br/>for <font color="Olive" title="b5">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K1" title="SFMASTR3:func.1">StepForUp</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">bb</font>,<font color="Olive" title="b3">cc</font>,<font color="Olive" title="b4">Ig</font>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Olive" title="b5">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">bb</font>,<font color="Olive" title="b3">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Olive" title="b4">Ig</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Olive" title="b5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Olive" title="b3">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Olive" title="b2">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )</span><br/><b>let </b><font color="Maroon" title="c2">a</font> be  <a href="sf_mastr.html#NV2" title="SF_MASTR:NV.2">read-write</a>  <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">bb</font>, <font color="Olive" title="b2">cc</font> being   <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> <br/> for <font color="Olive" title="b3">Ig</font> being <a href="scmfsa7b.html#V1" title="SCMFSA7B:attr.1">good</a> <a href="scmnorm.html#NM2" title="SCMNORM:NM.2">Program</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>   st <font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="sfmastr3.html#R1" title="SFMASTR3:pred.1">ProperForUpBody</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">bb</font>,<font color="Olive" title="b2">cc</font>,<font color="Olive" title="b3">Ig</font>,<font color="Maroon" title="c1">s</font> holds <br/>for <font color="Olive" title="b4">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K1" title="SFMASTR3:func.1">StepForUp</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">bb</font>,<font color="Olive" title="b2">cc</font>,<font color="Olive" title="b3">Ig</font>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">bb</font>,<font color="Olive" title="b2">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Olive" title="b3">Ig</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Olive" title="b4">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Olive" title="b2">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Olive" title="b1">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )</span><br/><b>let </b><font color="Maroon" title="c3">bb</font>, <font color="Maroon" title="c4">cc</font> be    <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Ig</font> being <a href="scmfsa7b.html#V1" title="SCMFSA7B:attr.1">good</a> <a href="scmnorm.html#NM2" title="SCMNORM:NM.2">Program</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>   st <font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="sfmastr3.html#R1" title="SFMASTR3:pred.1">ProperForUpBody</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Olive" title="b1">Ig</font>,<font color="Maroon" title="c1">s</font> holds <br/>for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K1" title="SFMASTR3:func.1">StepForUp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Olive" title="b1">Ig</font>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Olive" title="b1">Ig</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )</span><br/><b>let </b><font color="Maroon" title="c5">Ig</font> be  <a href="scmfsa7b.html#V1" title="SCMFSA7B:attr.1">good</a> <a href="scmnorm.html#NM2" title="SCMNORM:NM.2">Program</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="sfmastr3.html#R1" title="SFMASTR3:pred.1">ProperForUpBody</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Maroon" title="c5">Ig</font>,<font color="Maroon" title="c1">s</font> implies for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K1" title="SFMASTR3:func.1">StepForUp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Maroon" title="c5">Ig</font>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 ) )</span><br/>









<b>set </b><font color="Maroon" title="c6">D</font> = <a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> ;<br/>
<b>set </b><font color="Maroon" title="c7">I</font> = <font color="Maroon" title="c5">Ig</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green" title="E23">A2</font></i>: 
 <a href="sfmastr3.html#R1" title="SFMASTR3:pred.1">ProperForUpBody</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Maroon" title="c5">Ig</font>,<font color="Maroon" title="c1">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K1" title="SFMASTR3:func.1">StepForUp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Maroon" title="c5">Ig</font>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )</span><br/>

<b>set </b><font color="Maroon" title="c8">SF</font> =  <a href="sfmastr3.html#K1" title="SFMASTR3:func.1">StepForUp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Maroon" title="c5">Ig</font>,<font color="Maroon" title="c1">s</font>;<br/>
<b>set </b><font color="Maroon" title="c9">aux</font> = 1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p1">(<span class="default"><span class="p2"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">s2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa6b.html#K1" title="SCMFSA6B:func.1">+*</a> <span class="p2">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p3">(<span class="default"><span class="p4"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">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" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="scmfsa6b.html#K1" title="SCMFSA6B:func.1">+*</a> <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c11">IB</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">Ig</font> <a href="scmfsa6a.html#K7" title="SCMFSA6A:func.7">';'</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">AddTo</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K7" title="SCMFSA6A:func.7">';'</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K9" title="SCMFSA_2:func.9">SubFrom</a> <span class="p2">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p3">(<span class="default"><span class="p4"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c12">SW2</font> =  <a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Ig</font> <a href="scmfsa6a.html#K7" title="SCMFSA6A:func.7">';'</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">AddTo</a> <font color="Maroon" title="c2">a</font>,<span class="p4">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K7" title="SCMFSA6A:func.7">';'</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K9" title="SCMFSA_2:func.9">SubFrom</a> <span class="p3">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p4">(<span class="default"><span class="p5"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p5">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa6b.html#K1" title="SCMFSA6B:func.1">+*</a> <span class="p3">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p4">(<span class="default"><span class="p5"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p5">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">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" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="scmfsa6b.html#K1" title="SCMFSA6B:func.1">+*</a> <font color="Maroon" title="c2">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c13">scb1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K1" title="SFMASTR3:func.1">StepForUp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Maroon" title="c5">Ig</font>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> $1</span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 );<br/>
<a NAME="E3:23"/><i><font color="Green" title="E24">A3</font></i>: 
1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p1">(<span class="default"><span class="p2"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">s</font>
 
<b>by </b><i><a class="ref" href="scmfsa_2.html#T66" title="SCMFSA_2:th.66">SCMFSA_2:66</a></i>;<br/>
<a NAME="E4:23"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E5:23"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:23"/><b>then </b><i><font color="Green" title="E25">A4</font></i>: 
1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p1">(<span class="default"><span class="p2"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="sfmastr1.html#T21" title="SFMASTR1:th.21">SFMASTR1:21</a></i>;<br/>
<a NAME="E7:23"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green" title="E27">A6</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K1" title="SFMASTR3:func.1">StepForUp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Maroon" title="c5">Ig</font>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span><br/>

<a NAME="E2:23_1"/>
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <span class="p2">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p3">(<span class="default"><span class="p4"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Ig</font> <a href="scmfsa6a.html#K7" title="SCMFSA6A:func.7">';'</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">AddTo</a> <font color="Maroon" title="c2">a</font>,<span class="p5">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="scmfsa6a.html#K7" title="SCMFSA6A:func.7">';'</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K9" title="SCMFSA_2:func.9">SubFrom</a> <span class="p4">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p5">(<span class="default"><span class="p0"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p0">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa6b.html#K1" title="SCMFSA6B:func.1">+*</a> <span class="p4">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p5">(<span class="default"><span class="p0"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p0">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">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" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="scmfsa6b.html#K1" title="SCMFSA6B:func.1">+*</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa6b.html#K1" title="SCMFSA6B:func.1">+*</a> <span class="p2">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p3">(<span class="default"><span class="p4"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">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" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="scmfsa6b.html#K1" title="SCMFSA6B:func.1">+*</a> <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span>
 
<b>by </b><i><a class="ref" href="scmfsa_9.html#D5" title="SCMFSA_9:def.5">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#K1" title="SFMASTR3:func.1">StepForUp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Maroon" title="c5">Ig</font>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa6b.html#K1" title="SCMFSA6B:func.1">+*</a> <span class="p2">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p3">(<span class="default"><span class="p4"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">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" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sfmastr3.html#E6:23"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<b>by </b><i><a class="txt" href="sfmastr3.html#E3:23"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>
;<br/>
<b>hence </b><a NAME="E4:23_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="sfmastr3.html#E1:23_1"><i><font color="Green" title="E27">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:23"/><i><font color="Green" title="E27">A7</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:23_2"/><i><font color="Green" title="E28">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">k</font>]
 <b>and </b><br/><a NAME="E2:23_2"/><i><font color="Green" title="E29">A9</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K1" title="SFMASTR3:func.1">StepForUp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Maroon" title="c5">Ig</font>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span><br/>

<a NAME="E3:23_2"/><i><font color="Green" title="E30">A10</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K1" title="SFMASTR3:func.1">StepForUp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Maroon" title="c5">Ig</font>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_2_1"/><i><font color="Green" title="E31">A11</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K1" title="SFMASTR3:func.1">StepForUp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Maroon" title="c5">Ig</font>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:23_2_1"/><b>then </b>
 <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K1" title="SFMASTR3:func.1">StepForUp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Maroon" title="c5">Ig</font>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K1" title="SFMASTR3:func.1">StepForUp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Maroon" title="c5">Ig</font>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c14">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="scmfsa9a.html#T37" title="SCMFSA9A:th.37">SCMFSA9A:37</a></i>;<br/>
<b>hence </b><a NAME="E3:23_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sfmastr3.html#E2:23_2"><i><font color="Green" title="E29">A9</font></i></a>, <a class="txt" href="sfmastr3.html#E1:23_2_1"><i><font color="Green" title="E31">A11</font></i></a>, <a class="ref" href="scmfsa6a.html#T38" title="SCMFSA6A:th.38">SCMFSA6A:38</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c15">scb1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="sfmastr3.html#E1:23_2"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<a NAME="E5:23_2"/><i><font color="Green" title="E31">A12</font></i>: 
<font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">scb1</font>
 
<b>by </b><i><a class="txt" href="sfmastr3.html#E1:23_2"><i><font color="Green" title="E28">A8</font></i></a>, <a class="txt" href="sfmastr3.html#E3:23_2"><i><font color="Green" title="E30">A10</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>assume </b><a NAME="E6:23_2"/>
<font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E7:23_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#K1" title="SFMASTR3:func.1">StepForUp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Maroon" title="c5">Ig</font>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p2">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p3">(<span class="default"><span class="p4"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">scb1</font>
 
<b>by </b><i><a class="txt" href="sfmastr3.html#E2:23_2"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E8:23_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sfmastr3.html#E1:23"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="sfmastr3.html#E2:23"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="sfmastr3.html#E5:23_2"><i><font color="Green" title="E31">A12</font></i></a>, <a class="ref" href="sfmastr3.html#T25" target="_self" title="SFMASTR3:th.25">Th25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:23"/><i><font color="Green" title="E28">A13</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="sfmastr3.html#E7:23"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="sfmastr3.html#E8:23"><i><font color="Green" title="E27">A7</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c14">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K1" title="SFMASTR3:func.1">StepForUp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Maroon" title="c5">Ig</font>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )</span><br/>

<b>thus </b><a NAME="E10:23"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">k</font>]
 <b>by </b><i><a class="txt" href="sfmastr3.html#E9:23"><i><font color="Green" title="E28">A13</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K1" title="SFMASTR3:func.1">StepForUp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Maroon" title="c5">Ig</font>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<b>assume </b><a NAME="E11:23"/><i><font color="Green" title="E29">A14</font></i>: 
<font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K1" title="SFMASTR3:func.1">StepForUp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Maroon" title="c5">Ig</font>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E12:23"/><b>then </b><i><font color="Green" title="E30">A15</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sfmastr3.html#K1" title="SFMASTR3:func.1">StepForUp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Maroon" title="c5">Ig</font>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p2">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p3">(<span class="default"><span class="p4"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="sfmastr3.html#E1:23"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="sfmastr3.html#E2:23"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="sfmastr3.html#T25" target="_self" title="SFMASTR3:th.25">Th25</a></i>;<br/>
<a NAME="E13:23"/>
<font color="Maroon" title="c14">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">cc</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">k</font>
 
<b>by </b><i><a class="txt" href="sfmastr3.html#E11:23"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<b>hence </b><a NAME="E14:23"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr3.html#K1" title="SFMASTR3:func.1">StepForUp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font>,<font color="Maroon" title="c5">Ig</font>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <span class="p1">(<span class="default">1 <a href="sfmastr1.html#NK7" title="SFMASTR1:NK.7">-stRWNotIn</a> <span class="p2">(<span class="default"><span class="p3"><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">bb</font>,<font color="Maroon" title="c4">cc</font></span><a href="sfmastr1.html#K3" title="SFMASTR1:func.3">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="sf_mastr.html#K2" title="SF_MASTR:func.2">UsedIntLoc</a> <font color="Maroon" title="c5">Ig</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="sfmastr3.html#E12:23"><i><font color="Green" title="E30">A15</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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