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

<span class="kw">set </span><font color="Maroon" title="c6">I</font> =  <a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">s</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ami_1.html#D8" title="AMI_1:def.8">AMI_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span>,<font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">S</font> = <font color="Maroon" title="c7">s</font> as   <a href="scmpds_1.html#NM1" title="SCMPDS_1:NM.1">SCMPDS-State</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T155" title="PBOOLE:th.155">PBOOLE:155</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">Es</font> =  <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span>,<font color="Maroon" title="c7">s</font> as   <a href="scmpds_1.html#NM1" title="SCMPDS_1:NM.1">SCMPDS-State</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T155" title="PBOOLE:th.155">PBOOLE:155</a></span>;<br/>
<span class="lab"><font color="Green" title="E39">A1</font></span>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">Es</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>
<br/>.= 
 <a href="ami_2.html#K3" title="AMI_2:func.3">SCM-Memory</a> 

;<br/>
<div><span class="lab"><font color="Green" title="E40">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <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="c7">s</font> implies <font color="Maroon" title="c9">Es</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:79_2"/>
<font color="Maroon" title="c10">x</font> <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="c7">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">Es</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">m</font> = <font color="Maroon" title="c10">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K3" title="AMI_2:func.3">SCM-Memory</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:79_2_1"/>
( <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  or <font color="Maroon" title="c11">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a>  or <font color="Maroon" title="c11">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_1.html#T14" title="SCMPDS_1:th.14">SCMPDS_1:14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:79_2_1_1"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
<font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">Es</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c12">n</font> =  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c7">s</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  ;<br/><span class="kw">consider </span><font color="Maroon" title="c13">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:79_2_1_1"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">k</font>
 ;<br/><a NAME="E5:79_2_1_1"/><span class="lab"><font color="Green" title="E43">A6</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><a NAME="E6:79_2_1_1"/>
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c7">s</font> &amp;  <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> <font color="Maroon" title="c7">s</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_2.html#D20" target="_self" title="SCMPDS_2:def.20">Def20</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c9">Es</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> = 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c13">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_2_1_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E4:79_2_1_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="scmpds_2.html#T6" target="_self" title="SCMPDS_2:th.6">Th6</a>, <a class="ref" href="scmpds_2.html#T66" target="_self" title="SCMPDS_2:th.66">Th66</a></span>
<br/>.= 
<font color="Maroon" title="c13">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79_2_1_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E5:79_2_1_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_2_1_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E4:79_2_1_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:79_2_1_2"/>
<font color="Maroon" title="c11">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">Es</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">d</font> = <font color="Maroon" title="c10">x</font> as    <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_2.html#D2" target="_self" title="SCMPDS_2:def.2">Def2</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c9">Es</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> = 
<font color="Maroon" title="c7">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c12">d</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_2.html#T66" target="_self" title="SCMPDS_2:th.66">Th66</a></span>
<br/>.= 
<font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:79_2_1_3"/>
<font color="Maroon" title="c11">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">Es</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">v</font> = <font color="Maroon" title="c10">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">I0</font> =  <a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="scmpds_1.html#K1" title="SCMPDS_1:func.1">SCMPDS-Instr</a>  ;<br/> <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span>,<font color="Maroon" title="c7">s</font> = 
 <a href="scmpds_1.html#K24" title="SCMPDS_1:func.24">SCM-Exec-Res</a> <font color="Maroon" title="c13">I0</font>,<font color="Maroon" title="c8">S</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_1.html#D25" title="SCMPDS_1:def.25">SCMPDS_1:def 25</a></span>
<br/>.= 
 <a href="scmpds_1.html#K4" title="SCMPDS_1:func.4">SCM-Chg</a> <font color="Maroon" title="c8">S</font>,<span class="p1">(<span class="default"><a href="scmpds_1.html#K8" title="SCMPDS_1:func.8">jump_address</a> <font color="Maroon" title="c8">S</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">I0</font> <a href="scmpds_1.html#K11" title="SCMPDS_1:func.11">const_INT</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_1.html#D24" title="SCMPDS_1:def.24">SCMPDS_1:def 24</a></span>
;<br/><span class="kw">hence </span><font color="Maroon" title="c9">Es</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> = 
<font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_1.html#T28" title="SCMPDS_1:th.28">SCMPDS_1:28</a></span>
<br/>.= 
<font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>

;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">S</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>
<br/>.= 
 <a href="ami_2.html#K3" title="AMI_2:func.3">SCM-Memory</a> 

;<br/>
<span class="kw">hence </span><a NAME="E6:79"/>
 <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span>,<font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E3:79"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
