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

<b>let </b><font color="Maroon" title="c1">k</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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">R</font> being <a href="scmring1.html#V1" title="SCMRING1:attr.1">good</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a><br/> for <font color="Olive" title="b2">p</font> being  <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Olive" title="b1">R</font></span>)</span>  st not <font color="Olive" title="b1">R</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> holds <br/>for <font color="Olive" title="b3">F</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="ami_1.html#K14" title="AMI_1:func.14">FinPartSt</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Olive" title="b1">R</font></span>)</span>, <a href="ami_1.html#K14" title="AMI_1:func.14">FinPartSt</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Olive" title="b1">R</font></span>)</span>  st  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Olive" title="b1">R</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b3">F</font> is <a href="ami_1.html#V17" title="AMI_1:attr.17">data-only</a> holds <br/>( <font color="Olive" title="b2">p</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <font color="Olive" title="b3">F</font> iff  <a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Olive" title="b2">p</font>,<font color="Maroon" title="c1">k</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <font color="Olive" title="b3">F</font> )</span><br/><b>let </b><font color="Maroon" title="c2">R</font> be  <a href="scmring1.html#V1" title="SCMRING1:attr.1">good</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</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="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span>  st not <font color="Maroon" title="c2">R</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> holds <br/>for <font color="Olive" title="b2">F</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="ami_1.html#K14" title="AMI_1:func.14">FinPartSt</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span>, <a href="ami_1.html#K14" title="AMI_1:func.14">FinPartSt</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span>  st  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b2">F</font> is <a href="ami_1.html#V17" title="AMI_1:attr.17">data-only</a> holds <br/>( <font color="Olive" title="b1">p</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <font color="Olive" title="b2">F</font> iff  <a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">k</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <font color="Olive" title="b2">F</font> )</span><br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c2">R</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> implies for <font color="Olive" title="b1">F</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="ami_1.html#K14" title="AMI_1:func.14">FinPartSt</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span>, <a href="ami_1.html#K14" title="AMI_1:func.14">FinPartSt</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span>  st  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Olive" title="b1">F</font> is <a href="ami_1.html#V17" title="AMI_1:attr.17">data-only</a> holds <br/>( <font color="Maroon" title="c3">p</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <font color="Olive" title="b1">F</font> iff  <a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <font color="Olive" title="b1">F</font> ) )</span><br/>





<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E28">A1</font></i>: 
not <font color="Maroon" title="c2">R</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</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">F</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="ami_1.html#K14" title="AMI_1:func.14">FinPartSt</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span>, <a href="ami_1.html#K14" title="AMI_1:func.14">FinPartSt</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span>  st  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Olive" title="b1">F</font> is <a href="ami_1.html#V17" title="AMI_1:attr.17">data-only</a> holds <br/>( <font color="Maroon" title="c3">p</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <font color="Olive" title="b1">F</font> iff  <a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <font color="Olive" title="b1">F</font> )</span><br/>

<b>let </b><font color="Maroon" title="c4">F</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="ami_1.html#K14" title="AMI_1:func.14">FinPartSt</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span>, <a href="ami_1.html#K14" title="AMI_1:func.14">FinPartSt</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Maroon" title="c4">F</font> is <a href="ami_1.html#V17" title="AMI_1:attr.17">data-only</a> implies ( <font color="Maroon" title="c3">p</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <font color="Maroon" title="c4">F</font> iff  <a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <font color="Maroon" title="c4">F</font> ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:40"/><i><font color="Green" title="E29">A2</font></i>: 
 <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">p</font>
 <b>and </b><br/><a NAME="E3:40"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c4">F</font> is <a href="ami_1.html#V17" title="AMI_1:attr.17">data-only</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">p</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <font color="Maroon" title="c4">F</font> iff  <a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <font color="Maroon" title="c4">F</font> )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <font color="Maroon" title="c4">F</font> implies <font color="Maroon" title="c3">p</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <font color="Maroon" title="c4">F</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:40_1"/><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <font color="Maroon" title="c4">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <font color="Maroon" title="c4">F</font></span><br/><b>thus </b><a NAME="E2:40_1"/>
 <a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <font color="Maroon" title="c4">F</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ami_1.html#D29" title="AMI_1:def.29">AMI_1:def 29</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">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="c4">F</font> or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Olive">b<sub>1</sub></font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )</span><br/>

<b>assume </b><a NAME="E1:40_1_1"/><i><font color="Green" title="E32">A5</font></i>: 
<font color="Maroon" title="c5">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="c4">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Olive">b<sub>1</sub></font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )</span><br/>

<a NAME="E2:40_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K14" title="AMI_1:func.14">FinPartSt</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">s</font> = <font color="Maroon" title="c5">x</font> as   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> <b>by </b><i><a class="txt" href="scmring4.html#E1:40_1_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="ami_1.html#T125" title="AMI_1:th.125">AMI_1:125</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">s</font> = <font color="Maroon" title="c6">s</font> as  <a href="ami_1.html#V16" title="AMI_1:attr.16">data-only</a> <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> <b>by </b><i><a class="txt" href="scmring4.html#E3:40"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="scmring4.html#E1:40_1_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="ami_1.html#D51" title="AMI_1:def.51">AMI_1:def 51</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">s</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> &amp; <span class="p1">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E5:40_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font>
 ; <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="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font></span>)</span> )</span><br/>

<b>consider </b><font color="Maroon" title="c8">s1</font> being   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span><b> such that </b><br/><a NAME="E7:40_1_1"/><i><font color="Green" title="E33">A6</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s1</font> &amp; <font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c8">s1</font> is   <a href="ami_1.html#NM9" title="AMI_1:NM.9">pre-program</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c8">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="scmring4.html#E1:40_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="scmring4.html#E1:40_1_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="ami_1.html#D29" title="AMI_1:def.29">AMI_1:def 29</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">Fs1</font> = <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s1</font> as   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> <b>by </b><i><a class="txt" href="scmring4.html#E7:40_1_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="card_3.html#T80" title="CARD_3:th.80">CARD_3:80</a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<a NAME="E9:40_1_1"/><i><font color="Green" title="E34">A7</font></i>: 
<font color="Maroon" title="c9">Fs1</font> is <a href="ami_1.html#V16" title="AMI_1:attr.16">data-only</a>
 
<b>by </b><i><a class="txt" href="scmring4.html#E3:40"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="scmring4.html#E1:40_1_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="scmring4.html#E7:40_1_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="ami_1.html#D51" title="AMI_1:def.51">AMI_1:def 51</a></i>;<br/>
<a NAME="E10:40_1_1"/><b>then </b><i><font color="Green" title="E35">A8</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K35" title="AMI_1:func.35">DataPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c8">s1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="scmring4.html#E7:40_1_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="ami_1.html#T107" title="AMI_1:th.107">AMI_1:107</a></i>;<br/>
<a NAME="E11:40_1_1"/><i><font color="Green" title="E36">A9</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font></span>)</span>,<font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="txt" href="scmring4.html#E2:40"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="amistd_2.html#T78" title="AMISTD_2:th.78">AMISTD_2:78</a></i>;<br/>
<a NAME="E12:40_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E13:40_1_1"/><b>then </b><i><font color="Green" title="E37">A10</font></i>: 
 <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font></span>)</span>
 
<b>by </b><i><a class="txt" href="scmring4.html#E2:40"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E14:40_1_1"/>
<span class="p1">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font> is   <a href="ami_1.html#NM9" title="AMI_1:NM.9">pre-program</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmring4.html#E1:40"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="scmring4.html#E7:40_1_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="scmring4.html#E11:40_1_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="scmring4.html#T36" target="_self" title="SCMRING4:th.36">Th60</a>, <a class="ref" href="scmring4.html#T40" target="_self" title="SCMRING4:th.40">Th64</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font></span>)</span></span><br/>

 <a href="ami_1.html#K35" title="AMI_1:func.35">DataPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c8">s1</font></span>)</span></span>)</span> = 
 <a href="ami_1.html#K35" title="AMI_1:func.35">DataPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font></span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmring4.html#E1:40"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="scmring4.html#E7:40_1_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="scmring4.html#E13:40_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="ref" href="scmring4.html#T41" target="_self" title="SCMRING4:th.41">Th65</a></i>
<br/>.= 
 <a href="ami_1.html#K35" title="AMI_1:func.35">DataPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmring4.html#E2:40"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="amistd_2.html#T78" title="AMISTD_2:th.78">AMISTD_2:78</a></i>
;<br/>
<b>hence </b><a NAME="E16:40_1_1"/>
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmring4.html#E7:40_1_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="scmring4.html#E9:40_1_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="scmring4.html#E10:40_1_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="ami_1.html#T107" title="AMI_1:th.107">AMI_1:107</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>

<b>assume </b><a NAME="E5:40"/><i><font color="Green" title="E31">A11</font></i>: 
 <a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <font color="Maroon" title="c4">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="c3">p</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">computes</a> <font color="Maroon" title="c4">F</font></span><br/>

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ami_1.html#D29" title="AMI_1:def.29">AMI_1:def 29</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">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="c4">F</font> or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Olive">b<sub>1</sub></font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )</span><br/>

<b>assume </b><a NAME="E6:40"/><i><font color="Green" title="E32">A12</font></i>: 
<font color="Maroon" title="c5">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="c4">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Olive">b<sub>1</sub></font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )</span><br/>

<a NAME="E7:40"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K14" title="AMI_1:func.14">FinPartSt</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">s</font> = <font color="Maroon" title="c5">x</font> as   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> <b>by </b><i><a class="txt" href="scmring4.html#E6:40"><i><font color="Green" title="E32">A12</font></i></a>, <a class="ref" href="ami_1.html#T125" title="AMI_1:th.125">AMI_1:125</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">s</font> = <font color="Maroon" title="c6">s</font> as  <a href="ami_1.html#V16" title="AMI_1:attr.16">data-only</a> <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> <b>by </b><i><a class="txt" href="scmring4.html#E3:40"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="scmring4.html#E6:40"><i><font color="Green" title="E32">A12</font></i></a>, <a class="ref" href="ami_1.html#D51" title="AMI_1:def.51">AMI_1:def 51</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">s</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> &amp; <font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E10:40"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font></span>)</span> )</span><br/>

<b>consider </b><font color="Maroon" title="c8">s1</font> being   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span><b> such that </b><br/><a NAME="E12:40"/><i><font color="Green" title="E33">A13</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s1</font> &amp; <span class="p1">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c8">s1</font> is   <a href="ami_1.html#NM9" title="AMI_1:NM.9">pre-program</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c8">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="scmring4.html#E5:40"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="scmring4.html#E6:40"><i><font color="Green" title="E32">A12</font></i></a>, <a class="ref" href="ami_1.html#D29" title="AMI_1:def.29">AMI_1:def 29</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">Fs1</font> = <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s1</font> as   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> <b>by </b><i><a class="txt" href="scmring4.html#E12:40"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="card_3.html#T80" title="CARD_3:th.80">CARD_3:80</a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<a NAME="E14:40"/><i><font color="Green" title="E34">A14</font></i>: 
<font color="Maroon" title="c9">Fs1</font> is <a href="ami_1.html#V16" title="AMI_1:attr.16">data-only</a>
 
<b>by </b><i><a class="txt" href="scmring4.html#E3:40"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="scmring4.html#E6:40"><i><font color="Green" title="E32">A12</font></i></a>, <a class="txt" href="scmring4.html#E12:40"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="ami_1.html#D51" title="AMI_1:def.51">AMI_1:def 51</a></i>;<br/>
<a NAME="E15:40"/><b>then </b><i><font color="Green" title="E35">A15</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K35" title="AMI_1:func.35">DataPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c8">s1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="scmring4.html#E12:40"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="ami_1.html#T107" title="AMI_1:th.107">AMI_1:107</a></i>;<br/>
<a NAME="E16:40"/><i><font color="Green" title="E36">A16</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font></span>)</span>,<font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="txt" href="scmring4.html#E2:40"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="amistd_2.html#T78" title="AMISTD_2:th.78">AMISTD_2:78</a></i>;<br/>
<a NAME="E17:40"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E18:40"/><b>then </b><i><font color="Green" title="E37">A17</font></i>: 
 <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font></span>)</span>
 
<b>by </b><i><a class="txt" href="scmring4.html#E2:40"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E19:40"/><b>then </b><i><font color="Green" title="E38">A18</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font> is <a href="ami_1.html#V10" title="AMI_1:attr.10">autonomic</a>
 
<b>by </b><i><a class="txt" href="scmring4.html#E1:40"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="scmring4.html#E12:40"><i><font color="Green" title="E33">A13</font></i></a>, <a class="txt" href="scmring4.html#E16:40"><i><font color="Green" title="E36">A16</font></i></a>, <a class="ref" href="scmring4.html#T40" target="_self" title="SCMRING4:th.40">Th64</a></i>;<br/>
<b>hence </b><a NAME="E20:40"/>
<font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font> is   <a href="ami_1.html#NM9" title="AMI_1:NM.9">pre-program</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmring4.html#E1:40"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="scmring4.html#E12:40"><i><font color="Green" title="E33">A13</font></i></a>, <a class="txt" href="scmring4.html#E16:40"><i><font color="Green" title="E36">A16</font></i></a>, <a class="txt" href="scmring4.html#E18:40"><i><font color="Green" title="E37">A17</font></i></a>, <a class="ref" href="scmring4.html#T36" target="_self" title="SCMRING4:th.36">Th60</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font></span>)</span></span><br/>

<a NAME="E21:40"/><i><font color="Green" title="E39">A19</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font> is <a href="ami_1.html#V11" title="AMI_1:attr.11">halting</a>
 
<b>by </b><i><a class="txt" href="scmring4.html#E1:40"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="scmring4.html#E12:40"><i><font color="Green" title="E33">A13</font></i></a>, <a class="txt" href="scmring4.html#E16:40"><i><font color="Green" title="E36">A16</font></i></a>, <a class="txt" href="scmring4.html#E18:40"><i><font color="Green" title="E37">A17</font></i></a>, <a class="txt" href="scmring4.html#E19:40"><i><font color="Green" title="E38">A18</font></i></a>, <a class="ref" href="scmring4.html#T36" target="_self" title="SCMRING4:th.36">Th60</a></i>;<br/>
 <a href="ami_1.html#K35" title="AMI_1:func.35">DataPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c8">s1</font></span>)</span></span>)</span> = 
 <a href="ami_1.html#K35" title="AMI_1:func.35">DataPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font></span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmring4.html#E2:40"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="scmring4.html#E12:40"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="amistd_2.html#T78" title="AMISTD_2:th.78">AMISTD_2:78</a></i>
<br/>.= 
 <a href="ami_1.html#K35" title="AMI_1:func.35">DataPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmring4.html#E1:40"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="scmring4.html#E18:40"><i><font color="Green" title="E37">A17</font></i></a>, <a class="txt" href="scmring4.html#E19:40"><i><font color="Green" title="E38">A18</font></i></a>, <a class="txt" href="scmring4.html#E21:40"><i><font color="Green" title="E39">A19</font></i></a>, <a class="ref" href="scmring4.html#T41" target="_self" title="SCMRING4:th.41">Th65</a></i>
;<br/>
<b>hence </b><a NAME="E23:40"/>
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <font color="Maroon" title="c7">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmring4.html#E12:40"><i><font color="Green" title="E33">A13</font></i></a>, <a class="txt" href="scmring4.html#E14:40"><i><font color="Green" title="E34">A14</font></i></a>, <a class="txt" href="scmring4.html#E15:40"><i><font color="Green" title="E35">A15</font></i></a>, <a class="ref" href="ami_1.html#T107" title="AMI_1:th.107">AMI_1:107</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>
