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

<b>let </b><font color="Maroon" title="c4">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">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="c4">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="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">Ig</font>,<font color="Maroon" title="c4">s</font> &amp;  <a href="scmfsa9a.html#R4" title="SCMFSA9A:pred.4">WithVariantWhile&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">Ig</font>,<font color="Maroon" title="c4">s</font> holds <br/> ex <font color="Olive" title="b3">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b3">f</font> is <a href="scmfsa9a.html#V1" title="SCMFSA9A:attr.1">on_data_only</a> &amp; ( 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/>( <font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">Ig</font>,<font color="Maroon" title="c4">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> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">Ig</font>,<font color="Maroon" title="c4">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> <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) )</span><br/><b>let </b><font color="Maroon" title="c5">a</font> be  <a href="sf_mastr.html#NV2" title="SF_MASTR:NV.2">read-write</a>  <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> ; <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="c4">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="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">Ig</font>,<font color="Maroon" title="c4">s</font> &amp;  <a href="scmfsa9a.html#R4" title="SCMFSA9A:pred.4">WithVariantWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">Ig</font>,<font color="Maroon" title="c4">s</font> holds <br/> ex <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b2">f</font> is <a href="scmfsa9a.html#V1" title="SCMFSA9A:attr.1">on_data_only</a> &amp; ( for <font color="Olive" title="b3">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/>( <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Olive" title="b3">k</font></span>)</span> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) )</span><br/><b>let </b><font color="Maroon" title="c6">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="c4">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="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font> &amp;  <a href="scmfsa9a.html#R4" title="SCMFSA9A:pred.4">WithVariantWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font> implies  ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">f</font> is <a href="scmfsa9a.html#V1" title="SCMFSA9A:attr.1">on_data_only</a> &amp; ( 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/>( <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) ) )</span><br/>





<b>set </b><font color="Maroon" title="c7">I</font> = <font color="Maroon" title="c6">Ig</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:53"/><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon" title="c4">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="E2:53"/><i><font color="Green" title="E47">A2</font></i>: 
 <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font>
 <b>and </b><br/><a NAME="E3:53"/><i><font color="Green" title="E48">A3</font></i>: 
 <a href="scmfsa9a.html#R4" title="SCMFSA9A:pred.4">WithVariantWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">f</font> is <a href="scmfsa9a.html#V1" title="SCMFSA9A:attr.1">on_data_only</a> &amp; ( 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/>( <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) )</span><br/>

<b>set </b><font color="Maroon" title="c8">SW</font> =  <a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font>;<br/>
<b>consider </b><font color="Maroon" title="c9">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:53"/><i><font color="Green" title="E49">A4</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 <br/>( <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="scmfsa9a.html#E3:53"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="scmfsa9a.html#D5" target="_self" title="SCMFSA9A:def.5">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">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> <b> such that </b><br/><a NAME="E7:53"/><i><font color="Green" title="E50">A5</font></i>: 
 <a href="scmfsa9a.html#K2" title="SCMFSA9A:func.2">ExitsAtWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">K</font>
 <b>and </b><br/><a NAME="E8:53"/><i><font color="Green" title="E51">A6</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c10">K</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E9:53"/>
for <font color="Olive" title="b1">i</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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c10">K</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font>
 <b>and </b><br/><a NAME="E10:53"/>
 <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font></span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p4">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font></span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p5">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></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="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c10">K</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa9a.html#E2:53"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="scmfsa9a.html#E3:53"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="scmfsa9a.html#D6" target="_self" title="SCMFSA9A:def.6">Def6</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <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 href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>(  ex <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 <br/>( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font> &amp;  <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> $1 <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="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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> ) or ( ( 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/>(  not <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font> or  not  <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> $1 <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="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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> ) ) &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) );<br/>
<a NAME="E11:53"/><i><font color="Green" title="E52">A7</font></i>: 
for <font color="Olive" title="b1">x</font> being  <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>   ex <font color="Olive" title="b2">y</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">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</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">  ex <font color="Olive" title="b1">y</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="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1"/>
(  ex <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 <br/>( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font> &amp;  <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <font color="Maroon" title="c11">x</font> <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="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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> ) or 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/>(  not <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font> or  not  <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <font color="Maroon" title="c11">x</font> <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="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_1"/>
 ex <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 <br/>( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font> &amp;  <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <font color="Maroon" title="c11">x</font> <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="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</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="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">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> <b> such that </b><br/><a NAME="E3:53_1_1_1"/><i><font color="Green" title="E53">A8</font></i>: 
( <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font> &amp;  <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <font color="Maroon" title="c11">x</font> <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="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c12">k</font></span>)</span> )
 ;<br/><b>take </b>
<font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c12">k</font></span>)</span>
; <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="c11">x</font>,<font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c12">k</font></span>)</span>]</span><br/><b>thus </b><a NAME="E4:53_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c12">k</font></span>)</span>]
 <b>by </b><i><a class="txt" href="scmfsa9a.html#E3:53_1_1_1"><i><font color="Green" title="E53">A8</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_2"/><i><font color="Green" title="E53">A9</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 <br/>(  not <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font> or  not  <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <font color="Maroon" title="c11">x</font> <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="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</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="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><b>take </b>
 <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">S<sub>1</sub></font>[<font color="Maroon" title="c11">x</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]</span><br/><b>thus </b><a NAME="E2:53_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">x</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <b>by </b><i><a class="txt" href="scmfsa9a.html#E1:53_1_1_2"><i><font color="Green" title="E53">A9</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></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c11">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E13:53"/><i><font color="Green" title="E53">A10</font></i>: 
for <font color="Olive" title="b1">x</font> being  <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="scmfsa9a.html#E11:53"><i><font color="Green" title="E52">A7</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c11">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">f</font> is <a href="scmfsa9a.html#V1" title="SCMFSA9A:attr.1">on_data_only</a> &amp; ( 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/>( <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) )</span><br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="scmfsa9a.html#D7" target="_self" title="SCMFSA9A:def.7">SCMFSA9A:def 7</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">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/>( <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span>
<div class="add"><b>let </b><font color="Maroon" title="c12">s1</font>, <font color="Maroon" title="c13">s2</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"> (  <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <font color="Maroon" title="c12">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <font color="Maroon" title="c13">s2</font> implies <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">s2</font> )</span><br/><b>assume </b><a NAME="E1:53_2"/><i><font color="Green" title="E54">A11</font></i>: 
 <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <font color="Maroon" title="c12">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <font color="Maroon" title="c13">s2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">s2</font></span><br/><a NAME="E2:53_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">s1</font>,<font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">s1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">s2</font>,<font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">s2</font>] )
 <b>by </b><i><a class="txt" href="scmfsa9a.html#E13:53"><i><font color="Green" title="E53">A10</font></i></a></i>;<br/><b>hence </b><a NAME="E3:53_2"/>
<font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">s2</font>
 <b>by </b><i><a class="txt" href="scmfsa9a.html#E1:53"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="scmfsa9a.html#E2:53"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="scmfsa9a.html#E3:53"><i><font color="Green" title="E48">A3</font></i></a>, <a class="txt" href="scmfsa9a.html#E7:53"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="scmfsa9a.html#E1:53_2"><i><font color="Green" title="E54">A11</font></i></a>, <a class="ref" href="scmfsa9a.html#T45" target="_self" title="SCMFSA9A:th.45">Th45</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>let </b><font color="Maroon" title="c12">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" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c12">k</font></span>)</span> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_3"/>
( <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">K</font> or <font color="Maroon" title="c10">K</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_3_1"><b>suppose </b></a><a NAME="E1:53_3_1"/><i><font color="Green" title="E54">A12</font></i>: 
<font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c12">k</font></span>)</span> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><div class="add"><a NAME="E2:53_3_1"/><b>then </b><i><font color="Green" title="E55">A13</font></i>: 
<font color="Maroon" title="c12">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="c10">K</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">kk</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> <b> such that </b><br/><a NAME="E4:53_3_1"/><i><font color="Green" title="E56">A14</font></i>: 
( <font color="Maroon" title="c13">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font> &amp;  <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="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c12">k</font></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="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c13">kk</font></span>)</span> &amp; <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c13">kk</font></span>)</span> )
 <b>by </b><i><a class="txt" href="scmfsa9a.html#E13:53"><i><font color="Green" title="E53">A10</font></i></a>, <a class="txt" href="scmfsa9a.html#E1:53_3_1"><i><font color="Green" title="E54">A12</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c14">kk1</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> <b> such that </b><br/><a NAME="E6:53_3_1"/><i><font color="Green" title="E57">A15</font></i>: 
( <font color="Maroon" title="c14">kk1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font> &amp;  <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="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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="c12">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="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c14">kk1</font></span>)</span> &amp; <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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="c12">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> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c14">kk1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="scmfsa9a.html#E13:53"><i><font color="Green" title="E53">A10</font></i></a>, <a class="txt" href="scmfsa9a.html#E2:53_3_1"><i><font color="Green" title="E55">A13</font></i></a></i>;<br/><a NAME="E7:53_3_1"/>
( <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">kk</font> &amp; <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">kk1</font> )
 <b>by </b><i><a class="txt" href="scmfsa9a.html#E1:53"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="scmfsa9a.html#E2:53"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="scmfsa9a.html#E3:53"><i><font color="Green" title="E48">A3</font></i></a>, <a class="txt" href="scmfsa9a.html#E7:53"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="scmfsa9a.html#E1:53_3_1"><i><font color="Green" title="E54">A12</font></i></a>, <a class="txt" href="scmfsa9a.html#E2:53_3_1"><i><font color="Green" title="E55">A13</font></i></a>, <a class="txt" href="scmfsa9a.html#E4:53_3_1"><i><font color="Green" title="E56">A14</font></i></a>, <a class="txt" href="scmfsa9a.html#E6:53_3_1"><i><font color="Green" title="E57">A15</font></i></a>, <a class="ref" href="scmfsa9a.html#T45" target="_self" title="SCMFSA9A:th.45">Th45</a></i>;<br/><b>hence </b><a NAME="E8:53_3_1"/>
( <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c12">k</font></span>)</span> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="scmfsa9a.html#E5:53"><i><font color="Green" title="E49">A4</font></i></a>, <a class="txt" href="scmfsa9a.html#E4:53_3_1"><i><font color="Green" title="E56">A14</font></i></a>, <a class="txt" href="scmfsa9a.html#E6:53_3_1"><i><font color="Green" title="E57">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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_3_2"><b>suppose </b></a><a NAME="E1:53_3_2"/>
<font color="Maroon" title="c10">K</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c12">k</font></span>)</span> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><div class="add"><a NAME="E2:53_3_2"/><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="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c10">K</font></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="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa9a.html#E8:53"><i><font color="Green" title="E51">A6</font></i></a>, <a class="ref" href="scmfsa9a.html#T43" target="_self" title="SCMFSA9A:th.43">Th43</a></i>;<br/><b>hence </b><a NAME="E3:53_3_2"/>
( <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">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="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c12">k</font></span>)</span> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="scmfsa9a.html#E8:53"><i><font color="Green" title="E51">A6</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></div><b>end;</b></div>

</div>
