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

<b>let </b><font color="Maroon" title="c1">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</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">p</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ,the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  holds   <a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <span class="p1">(<span class="default"><a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c1">i</font></span><br/><b>let </b><font color="Maroon" title="c2">p</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ,the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <span class="p1">(<span class="default"><a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c1">i</font></span><br/>



<a NAME="E1:13"/><i><font color="Green" title="E14">A1</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <span class="p2">(<span class="default"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="reloc.html#D5" target="_self" title="RELOC:def.5">Def5</a></i>;<br/>
<a NAME="E2:13"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="reloc.html#D5" target="_self" title="RELOC:def.5">Def5</a></i>;<br/>
<b>then </b><i><font color="Green" title="E15">A2</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> = 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">i</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  : <font color="Olive" title="b1">m</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> <span class="p1">(<span class="default"><a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> </span>}</span> 
<b>by </b><i><a class="ref" href="valued_1.html#D12" title="VALUED_1:def.12">VALUED_1:def 12</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <span class="p2">(<span class="default"><a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#D12" title="VALUED_1:def.12">VALUED_1:def 12</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <span class="p2">(<span class="default"><a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span> implies <span class="p1">(<span class="default"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <span class="p2">(<span class="default"><a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <span class="p2">(<span class="default"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> )</span><br/><a NAME="E1:13_2"/><i><font color="Green" title="E16">A3</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <span class="p2">(<span class="default"><a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><b>assume </b><a NAME="E2:13_2"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <span class="p2">(<span class="default"><a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span>
 ; <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"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <span class="p2">(<span class="default"><a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <span class="p2">(<span class="default"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span><br/><b>then reconsider </b><font color="Maroon" title="c4">x'</font> = <font color="Maroon" title="c3">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E1:13_2"><i><font color="Green" title="E16">A3</font></i></a></i>;<br/><a NAME="E4:13_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">i</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  : <font color="Olive" title="b1">m</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> <span class="p1">(<span class="default"><a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E2:13_2"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="valued_1.html#D12" title="VALUED_1:def.12">VALUED_1:def 12</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:13_2"/><i><font color="Green" title="E18">A5</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c5">m</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> <span class="p1">(<span class="default"><a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> )
 ;<br/><a NAME="E7:13_2"/><i><font color="Green" title="E19">A7</font></i>: 
 <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <font color="Maroon" title="c5">m</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="c2">p</font>
 <b>by </b><i><a class="txt" href="#E6:13_2"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="reloc.html#D5" target="_self" title="RELOC:def.5">Def5</a></i>;<br/><a NAME="E8:13_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">mm</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">i</font></span>)</span> where <font color="Olive" title="b1">mm</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  : <font color="Olive" title="b1">mm</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="c2">p</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="valued_1.html#D12" title="VALUED_1:def.12">VALUED_1:def 12</a></i>;<br/><a NAME="E9:13_2"/><b>then </b><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:13_2"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="#E7:13_2"><i><font color="Green" title="E19">A7</font></i></a></i>;<br/><i><font color="Green" title="E21">A9</font></i>: <font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">m</font> = 
<font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font>
<b>by </b><i><a class="txt" href="#E7:13_2"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:13_2"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="valued_1.html#D12" title="VALUED_1:def.12">VALUED_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x'</font>
<b>by </b><i><a class="txt" href="#E6:13_2"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="#E9:13_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <span class="p2">(<span class="default"><a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> = 
<span class="p1">(<span class="default"><a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font>
<b>by </b><i><a class="txt" href="#E6:13_2"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="valued_1.html#D12" title="VALUED_1:def.12">VALUED_1:def 12</a></i>
<br/>.= 
 <a href="reloc.html#K1" title="RELOC:func.1">IncAddr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x'</font></span>)</span>,<font color="Maroon" title="c1">i</font>
<b>by </b><i><a class="txt" href="#E7:13_2"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="#E10:13_2"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="reloc.html#D5" target="_self" title="RELOC:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <span class="p2">(<span class="default"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="txt" href="#E9:13_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="reloc.html#D5" target="_self" title="RELOC:def.5">Def5</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>hence </b><a NAME="E5:13"/>
 <a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <span class="p1">(<span class="default"><a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="reloc.html#K2" title="RELOC:func.2">IncAddr</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K5" title="SCMNORM:func.5">Shift</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c1">i</font>
 <b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E3:13"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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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