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<div class="add">

<span class="kw">thus </span><a NAME="E1:109_1_1"/>
 <a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font> is  <a href="amistd_2.html#V9" title="AMISTD_2:attr.9">IC-good</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font> is  <a href="amistd_2.html#V11" title="AMISTD_2:attr.11">Exec-preserving</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">I</font> be   <a href="ami_1.html#NM2" title="AMI_1:NM.2">Instruction</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="amistd_2.html#D18" title="AMISTD_2:def.18">AMISTD_2:def 18</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="c2">I</font> is  <a href="amistd_2.html#V8" title="AMISTD_2:attr.8">IC-good</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_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="E1:109_1_1_1_1"/>
( <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="scmring2.html#K4" title="SCMRING2:func.4">:=</a> <font color="Olive" title="b2">b</font> or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K5" title="SCMRING2:func.5">AddTo</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K6" title="SCMRING2:func.6">SubFrom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K7" title="SCMRING2:func.7">MultBy</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> or  ex <font color="Olive" title="b1">i1</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>  st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K9" title="SCMRING2:func.9">goto</a> <font color="Olive" title="b1">i1</font>,<font color="Maroon" title="c1">R</font> or  ex <font color="Olive" title="b1">a</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> ex <font color="Olive" title="b2">i1</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>  st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <font color="Olive" title="b2">i1</font> or  ex <font color="Olive" title="b1">a</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> ex <font color="Olive" title="b2">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="scmring2.html#K8" title="SCMRING2:func.8">:=</a> <font color="Olive" title="b2">r</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="scmring2.html#T8" title="SCMRING2:th.8">SCMRING2:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_1_1_1"/>
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <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="c2">I</font> is  <a href="amistd_2.html#V8" title="AMISTD_2:attr.8">IC-good</a> </span><br/><div class="add"><a NAME="E2:109_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmring2.html#T30" title="SCMRING2:th.30">SCMRING2:30</a></span>;<br/><span class="kw">hence </span><a NAME="E3:109_1_1_1_1_1"/>
<font color="Maroon" title="c2">I</font> is  <a href="amistd_2.html#V8" title="AMISTD_2:attr.8">IC-good</a> 
 ; <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="109_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="scmring2.html#K4" title="SCMRING2:func.4">:=</a> <font color="Olive" title="b2">b</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="c2">I</font> is  <a href="amistd_2.html#V8" title="AMISTD_2:attr.8">IC-good</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:109_1_1_1_1_2"/>
<font color="Maroon" title="c2">I</font> is  <a href="amistd_2.html#V8" title="AMISTD_2:attr.8">IC-good</a> 
 ; <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="109_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_1_1_3"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K5" title="SCMRING2:func.5">AddTo</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</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="c2">I</font> is  <a href="amistd_2.html#V8" title="AMISTD_2:attr.8">IC-good</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:109_1_1_1_1_3"/>
<font color="Maroon" title="c2">I</font> is  <a href="amistd_2.html#V8" title="AMISTD_2:attr.8">IC-good</a> 
 ; <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="109_1_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_1_1_4"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K6" title="SCMRING2:func.6">SubFrom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</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="c2">I</font> is  <a href="amistd_2.html#V8" title="AMISTD_2:attr.8">IC-good</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:109_1_1_1_1_4"/>
<font color="Maroon" title="c2">I</font> is  <a href="amistd_2.html#V8" title="AMISTD_2:attr.8">IC-good</a> 
 ; <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="109_1_1_1_1_5"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_1_1_5"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K7" title="SCMRING2:func.7">MultBy</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</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="c2">I</font> is  <a href="amistd_2.html#V8" title="AMISTD_2:attr.8">IC-good</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:109_1_1_1_1_5"/>
<font color="Maroon" title="c2">I</font> is  <a href="amistd_2.html#V8" title="AMISTD_2:attr.8">IC-good</a> 
 ; <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="109_1_1_1_1_6"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_1_1_6"/><span class="lab"><font color="Green" title="E66">A1</font></span>: 
 ex <font color="Olive" title="b1">i1</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>  st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K9" title="SCMRING2:func.9">goto</a> <font color="Olive" title="b1">i1</font>,<font color="Maroon" title="c1">R</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="c2">I</font> is  <a href="amistd_2.html#V8" title="AMISTD_2:attr.8">IC-good</a> </span><br/><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">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> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="amistd_2.html#D17" title="AMISTD_2:def.17">AMISTD_2:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c3">k</font>,<span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> or <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">IncAddr</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">k</font></span>)</span>,<font color="Olive">b<sub>2</sub></font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">s1</font>, <font color="Maroon" title="c5">s2</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c3">k</font>,<span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> or <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s1</font></span>)</span></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">IncAddr</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c5">s2</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E2:109_1_1_1_1_6"/>
<font color="Maroon" title="c5">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c3">k</font>,<span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s1</font></span>)</span></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">IncAddr</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c5">s2</font></span>)</span></span><br/><span class="kw">consider </span><font color="Maroon" title="c6">i1</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> <span class="kw"> such that </span><br/><a NAME="E4:109_1_1_1_1_6"/><span class="lab"><font color="Green" title="E67">A2</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K9" title="SCMRING2:func.9">goto</a> <font color="Maroon" title="c6">i1</font>,<font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:109_1_1_1_1_6"><span class="lab"><font color="Green" title="E66">A1</font></span></a></span>;<br/><a NAME="E5:109_1_1_1_1_6"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109_1_1_1_1_6"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="ref" href="scmring2.html#T17" title="SCMRING2:th.17">SCMRING2:17</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s1</font></span>)</span></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> = 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="scmring2.html#K9" title="SCMRING2:func.9">goto</a> <span class="p3">(<span class="default"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <span class="p4">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Maroon" title="c6">i1</font>,<span class="p0">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c5">s2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmring2.html#T17" title="SCMRING2:th.17">SCMRING2:17</a></span>
<br/>.= 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">IncAddr</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c5">s2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109_1_1_1_1_6"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="ref" href="scmring3.html#T69" target="_self" title="SCMRING3:th.69">Th69</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="109_1_1_1_1_7"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_1_1_7"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> ex <font color="Olive" title="b2">i1</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>  st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <font color="Olive" title="b2">i1</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="c2">I</font> is  <a href="amistd_2.html#V8" title="AMISTD_2:attr.8">IC-good</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">a</font> being    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c4">i1</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> <span class="kw"> such that </span><br/><a NAME="E3:109_1_1_1_1_7"/><span class="lab"><font color="Green" title="E66">A3</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <font color="Maroon" title="c4">i1</font>
 ;<br/><span class="kw">let </span><font color="Maroon" title="c5">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> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="amistd_2.html#D17" title="AMISTD_2:def.17">AMISTD_2:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c5">k</font>,<span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> or <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c5">k</font>,<span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">IncAddr</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c5">k</font></span>)</span>,<font color="Olive">b<sub>2</sub></font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">s1</font>, <font color="Maroon" title="c7">s2</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c5">k</font>,<span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> or <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c6">s1</font></span>)</span></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c5">k</font>,<span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">IncAddr</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c5">k</font></span>)</span>,<font color="Maroon" title="c7">s2</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E4:109_1_1_1_1_7"/><span class="lab"><font color="Green" title="E67">A4</font></span>: 
<font color="Maroon" title="c7">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c5">k</font>,<span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c6">s1</font></span>)</span></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c5">k</font>,<span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">IncAddr</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c5">k</font></span>)</span>,<font color="Maroon" title="c7">s2</font></span>)</span></span><br/><a NAME="E5:109_1_1_1_1_7"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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="c1">R</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c5">k</font>,<span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmring3.html#T3" target="_self" title="SCMRING3:th.3">Th3</a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/><a NAME="E6:109_1_1_1_1_7"/><span class="kw">then </span>
not <font color="Maroon" title="c3">a</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"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c5">k</font>,<span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E7:109_1_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A5</font></span>: 
<font color="Maroon" title="c6">s1</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s2</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109_1_1_1_1_7"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1_1_1_7_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_1_7_1_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="E1:109_1_1_1_1_7_1_1"/>
( <font color="Maroon" title="c6">s1</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> or <font color="Maroon" title="c6">s1</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_1_7_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_1_1_7_1_1_1"/><span class="lab"><font color="Green" title="E69">A6</font></span>: 
<font color="Maroon" title="c6">s1</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c6">s1</font></span>)</span></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c5">k</font>,<span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">IncAddr</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c5">k</font></span>)</span>,<font color="Maroon" title="c7">s2</font></span>)</span></span><br/><div class="add"><a NAME="E2:109_1_1_1_1_7_1_1_1"/><span class="kw">then </span>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c6">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_1_1_7"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="ref" href="scmring2.html#T18" title="SCMRING2:th.18">SCMRING2:18</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c6">s1</font></span>)</span></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c5">k</font>,<span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> = 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <span class="p3">(<span class="default"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <span class="p4">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Maroon" title="c4">i1</font>,<span class="p0">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c7">s2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:109_1_1_1_1_7"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="txt" href="#E1:109_1_1_1_1_7_1_1_1"><span class="lab"><font color="Green" title="E69">A6</font></span></a>, <a class="ref" href="scmring2.html#T18" title="SCMRING2:th.18">SCMRING2:18</a></span>
<br/>.= 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">IncAddr</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c5">k</font></span>)</span>,<font color="Maroon" title="c7">s2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_1_1_7"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="ref" href="scmring3.html#T70" target="_self" title="SCMRING3:th.70">Th70</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="109_1_1_1_1_7_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_1_1_7_1_1_2"/><span class="lab"><font color="Green" title="E69">A7</font></span>: 
<font color="Maroon" title="c6">s1</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c6">s1</font></span>)</span></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c5">k</font>,<span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">IncAddr</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c5">k</font></span>)</span>,<font color="Maroon" title="c7">s2</font></span>)</span></span><br/><div class="add"><a NAME="E2:109_1_1_1_1_7_1_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c5">k</font>,<span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/><a NAME="E3:109_1_1_1_1_7_1_1_2"/><span class="kw">then </span>
 <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="c1">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"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c5">k</font>,<span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A8</font></span>:  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c7">s2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c5">k</font>,<span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p2">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109_1_1_1_1_7"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>
<br/>.= 
 <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <span class="p3">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s1</font></span>)</span>,<span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></span>
;<br/><span class="lab"><font color="Green" title="E71">A9</font></span>:  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c6">s1</font></span>)</span> = 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_1_1_7"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="txt" href="#E1:109_1_1_1_1_7_1_1_2"><span class="lab"><font color="Green" title="E69">A7</font></span></a>, <a class="ref" href="scmring2.html#T18" title="SCMRING2:th.18">SCMRING2:18</a></span>
<br/>.= 
 <a href="amistd_1.html#K9" title="AMISTD_1:func.9">NextLoc</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s1</font></span>)</span>,<span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmring3.html#T68" target="_self" title="SCMRING3:th.68">Th68</a></span>
<br/>.= 
 <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <span class="p3">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s1</font></span>)</span>,<span class="p3">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

;<br/> <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c7">s2</font></span>)</span> = 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c7">s2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_1_1_7"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="txt" href="#E7:109_1_1_1_1_7"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="txt" href="#E1:109_1_1_1_1_7_1_1_2"><span class="lab"><font color="Green" title="E69">A7</font></span></a>, <a class="ref" href="scmring2.html#T18" title="SCMRING2:th.18">SCMRING2:18</a></span>
<br/>.= 
 <a href="amistd_1.html#K9" title="AMISTD_1:func.9">NextLoc</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c7">s2</font></span>)</span>,<span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmring3.html#T68" target="_self" title="SCMRING3:th.68">Th68</a></span>
<br/>.= 
 <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <span class="p4">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s1</font></span>)</span>,<span class="p4">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109_1_1_1_1_7_1_1_2"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="ref" href="amistd_1.html#D13" title="AMISTD_1:def.13">AMISTD_1:def 13</a></span>
<br/>.= 
 <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <span class="p4">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s1</font></span>)</span>,<span class="p4">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span>

;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c6">s1</font></span>)</span></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c5">k</font>,<span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c7">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p2">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:109_1_1_1_1_7_1_1_2"><span class="lab"><font color="Green" title="E71">A9</font></span></a>, <a class="ref" href="amistd_1.html#D13" title="AMISTD_1:def.13">AMISTD_1:def 13</a></span>
<br/>.= 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c7">s2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_1_1_7"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="txt" href="#E7:109_1_1_1_1_7"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="txt" href="#E1:109_1_1_1_1_7_1_1_2"><span class="lab"><font color="Green" title="E69">A7</font></span></a>, <a class="ref" href="scmring2.html#T18" title="SCMRING2:th.18">SCMRING2:18</a></span>
<br/>.= 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <span class="p3">(<span class="default"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <span class="p4">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Maroon" title="c4">i1</font>,<span class="p0">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c7">s2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:109_1_1_1_1_7"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="txt" href="#E1:109_1_1_1_1_7_1_1_2"><span class="lab"><font color="Green" title="E69">A7</font></span></a>, <a class="ref" href="scmring2.html#T18" title="SCMRING2:th.18">SCMRING2:18</a></span>
<br/>.= 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">IncAddr</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c5">k</font></span>)</span>,<font color="Maroon" title="c7">s2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_1_1_7"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="ref" href="scmring3.html#T70" target="_self" title="SCMRING3:th.70">Th70</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E9:109_1_1_1_1_7"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c6">s1</font></span>)</span></span>)</span> <a href="amistd_1.html#K8" title="AMISTD_1:func.8">+</a> <font color="Maroon" title="c5">k</font>,<span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" title="AMISTD_2:func.7">IncAddr</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c5">k</font></span>)</span>,<font color="Maroon" title="c7">s2</font></span>)</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_1_8"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_1_1_8"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> ex <font color="Olive" title="b2">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="scmring2.html#K8" title="SCMRING2:func.8">:=</a> <font color="Olive" title="b2">r</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="c2">I</font> is  <a href="amistd_2.html#V8" title="AMISTD_2:attr.8">IC-good</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:109_1_1_1_1_8"/>
<font color="Maroon" title="c2">I</font> is  <a href="amistd_2.html#V8" title="AMISTD_2:attr.8">IC-good</a> 
 ; <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>

<span class="kw">let </span><font color="Maroon" title="c2">I</font> be   <a href="ami_1.html#NM2" title="AMI_1:NM.2">Instruction</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="amistd_2.html#D20" title="AMISTD_2:def.20">AMISTD_2:def 20</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="c2">I</font> is  <a href="amistd_2.html#V10" title="AMISTD_2:attr.10">Exec-preserving</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">s1</font>, <font color="Maroon" title="c4">s2</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="amistd_2.html#D19" title="AMISTD_2:def.19">AMISTD_2:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  or  <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font>, <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E2:109_1_1"/><span class="lab"><font color="Green" title="E66">A11</font></span>: 
<font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</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">  <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font>, <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<a NAME="E3:109_1_1"/><span class="lab"><font color="Green" title="E67">A12</font></span>: 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c3">s1</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="c4">s2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="ref" href="ami_1.html#T121" title="AMI_1:th.121">AMI_1:121</a></span>;<br/>
<a NAME="E4:109_1_1"/><span class="lab"><font color="Green" title="E68">A13</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>
 
<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 NAME="E5:109_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A14</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</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="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span>
 
<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="109_1_1_2"><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="E1:109_1_1_2"/>
( <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="scmring2.html#K4" title="SCMRING2:func.4">:=</a> <font color="Olive" title="b2">b</font> or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K5" title="SCMRING2:func.5">AddTo</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K6" title="SCMRING2:func.6">SubFrom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K7" title="SCMRING2:func.7">MultBy</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> or  ex <font color="Olive" title="b1">i1</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>  st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K9" title="SCMRING2:func.9">goto</a> <font color="Olive" title="b1">i1</font>,<font color="Maroon" title="c1">R</font> or  ex <font color="Olive" title="b1">a</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> ex <font color="Olive" title="b2">i1</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>  st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <font color="Olive" title="b2">i1</font> or  ex <font color="Olive" title="b1">a</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> ex <font color="Olive" title="b2">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="scmring2.html#K8" title="SCMRING2:func.8">:=</a> <font color="Olive" title="b2">r</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="scmring2.html#T8" title="SCMRING2:th.8">SCMRING2:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_1"/>
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <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> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font>, <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/><div class="add"><a NAME="E2:109_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmring2.html#T30" title="SCMRING2:th.30">SCMRING2:30</a></span>;<br/><span class="kw">hence </span><a NAME="E3:109_1_1_2_1"/>
 <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font>, <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="ref" href="amistd_2.html#D19" title="AMISTD_2:def.19">AMISTD_2:def 19</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_2"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="scmring2.html#K4" title="SCMRING2:func.4">:=</a> <font color="Olive" title="b2">b</font>
 ; <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> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font>, <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E3:109_1_1_2_2"/><span class="lab"><font color="Green" title="E70">A15</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="scmring2.html#K4" title="SCMRING2:func.4">:=</a> <font color="Maroon" title="c6">b</font>
 ;<br/><a NAME="E4:109_1_1_2_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  holds <br/><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  implies <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:109_1_1_2_2_1"/><span class="lab"><font color="Green" title="E71">A16</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/>

<a NAME="E2:109_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A17</font></span>: 
not <font color="Maroon" title="c7">x</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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_2_2_1_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="E1:109_1_1_2_2_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">R</font></span>)</span> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> or ( <font color="Maroon" title="c7">x</font> is    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109_1_1"><span class="lab"><font color="Green" title="E68">A13</font></span></a>, <a class="txt" href="#E1:109_1_1_2_2_1"><span class="lab"><font color="Green" title="E71">A16</font></span></a>, <a class="txt" href="#E2:109_1_1_2_2_1"><span class="lab"><font color="Green" title="E72">A17</font></span></a>, <a class="ref" href="scmring3.html#T5" target="_self" title="SCMRING3:th.5">Th5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E73">A18</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">R</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c3">s1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_2"><span class="lab"><font color="Green" title="E70">A15</font></span></a>, <a class="ref" href="scmring2.html#T13" title="SCMRING2:th.13">SCMRING2:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1"><span class="lab"><font color="Green" title="E67">A12</font></span></a>, <a class="txt" href="#E3:109_1_1_2_2"><span class="lab"><font color="Green" title="E70">A15</font></span></a>, <a class="txt" href="#E1:109_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E73">A18</font></span></a>, <a class="ref" href="scmring2.html#T13" title="SCMRING2:th.13">SCMRING2:13</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="109_1_1_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E73">A19</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
<font color="Maroon" title="c3">s1</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c6">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_2"><span class="lab"><font color="Green" title="E70">A15</font></span></a>, <a class="ref" href="scmring2.html#T13" title="SCMRING2:th.13">SCMRING2:13</a></span>
<br/>.= 
<font color="Maroon" title="c4">s2</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c6">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="ref" href="scmring3.html#T7" target="_self" title="SCMRING3:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_2"><span class="lab"><font color="Green" title="E70">A15</font></span></a>, <a class="txt" href="#E1:109_1_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E73">A19</font></span></a>, <a class="ref" href="scmring2.html#T13" title="SCMRING2:th.13">SCMRING2:13</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="109_1_1_2_2_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:109_1_1_2_2_1_1_3"/><span class="lab"><font color="Green" title="E73">A20</font></span>: 
<font color="Maroon" title="c7">x</font> is    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font>
 <span class="kw">and </span><br/><a NAME="E2:109_1_1_2_2_1_1_3"/><span class="lab"><font color="Green" title="E74">A21</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><div class="add"><span class="kw">thus </span><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_2"><span class="lab"><font color="Green" title="E70">A15</font></span></a>, <a class="txt" href="#E1:109_1_1_2_2_1_1_3"><span class="lab"><font color="Green" title="E73">A20</font></span></a>, <a class="txt" href="#E2:109_1_1_2_2_1_1_3"><span class="lab"><font color="Green" title="E74">A21</font></span></a>, <a class="ref" href="scmring2.html#T13" title="SCMRING2:th.13">SCMRING2:13</a></span>
<br/>.= 
<font color="Maroon" title="c4">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="txt" href="#E1:109_1_1_2_2_1_1_3"><span class="lab"><font color="Green" title="E73">A20</font></span></a>, <a class="ref" href="scmring3.html#T7" target="_self" title="SCMRING3:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_2"><span class="lab"><font color="Green" title="E70">A15</font></span></a>, <a class="txt" href="#E1:109_1_1_2_2_1_1_3"><span class="lab"><font color="Green" title="E73">A20</font></span></a>, <a class="txt" href="#E2:109_1_1_2_2_1_1_3"><span class="lab"><font color="Green" title="E74">A21</font></span></a>, <a class="ref" href="scmring2.html#T13" title="SCMRING2:th.13">SCMRING2:13</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:109_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:109_1_1"><span class="lab"><font color="Green" title="E69">A14</font></span></a>, <a class="ref" href="funct_1.html#T165" title="FUNCT_1:th.165">FUNCT_1:165</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_7.html#D2" title="FUNCT_7:def.2">FUNCT_7:def 2</a> <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="109_1_1_2_3"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_3"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K5" title="SCMRING2:func.5">AddTo</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>
 ; <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> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font>, <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E3:109_1_1_2_3"/><span class="lab"><font color="Green" title="E70">A22</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K5" title="SCMRING2:func.5">AddTo</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>
 ;<br/><a NAME="E4:109_1_1_2_3"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  holds <br/><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1_2_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  implies <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:109_1_1_2_3_1"/><span class="lab"><font color="Green" title="E71">A23</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/>

<a NAME="E2:109_1_1_2_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A24</font></span>: 
not <font color="Maroon" title="c7">x</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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_2_3_1_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="E1:109_1_1_2_3_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">R</font></span>)</span> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> or ( <font color="Maroon" title="c7">x</font> is    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109_1_1"><span class="lab"><font color="Green" title="E68">A13</font></span></a>, <a class="txt" href="#E1:109_1_1_2_3_1"><span class="lab"><font color="Green" title="E71">A23</font></span></a>, <a class="txt" href="#E2:109_1_1_2_3_1"><span class="lab"><font color="Green" title="E72">A24</font></span></a>, <a class="ref" href="scmring3.html#T5" target="_self" title="SCMRING3:th.5">Th5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_2_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_3_1_1_1"/><span class="lab"><font color="Green" title="E73">A25</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">R</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c3">s1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_3"><span class="lab"><font color="Green" title="E70">A22</font></span></a>, <a class="ref" href="scmring2.html#T14" title="SCMRING2:th.14">SCMRING2:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1"><span class="lab"><font color="Green" title="E67">A12</font></span></a>, <a class="txt" href="#E3:109_1_1_2_3"><span class="lab"><font color="Green" title="E70">A22</font></span></a>, <a class="txt" href="#E1:109_1_1_2_3_1_1_1"><span class="lab"><font color="Green" title="E73">A25</font></span></a>, <a class="ref" href="scmring2.html#T14" title="SCMRING2:th.14">SCMRING2:14</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="109_1_1_2_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_3_1_1_2"/><span class="lab"><font color="Green" title="E73">A26</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_3"><span class="lab"><font color="Green" title="E70">A22</font></span></a>, <a class="ref" href="scmring2.html#T14" title="SCMRING2:th.14">SCMRING2:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s2</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="ref" href="scmring3.html#T7" target="_self" title="SCMRING3:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s2</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s2</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="ref" href="scmring3.html#T7" target="_self" title="SCMRING3:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_3"><span class="lab"><font color="Green" title="E70">A22</font></span></a>, <a class="txt" href="#E1:109_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E73">A26</font></span></a>, <a class="ref" href="scmring2.html#T14" title="SCMRING2:th.14">SCMRING2:14</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="109_1_1_2_3_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:109_1_1_2_3_1_1_3"/><span class="lab"><font color="Green" title="E73">A27</font></span>: 
<font color="Maroon" title="c7">x</font> is    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font>
 <span class="kw">and </span><br/><a NAME="E2:109_1_1_2_3_1_1_3"/><span class="lab"><font color="Green" title="E74">A28</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><div class="add"><span class="kw">thus </span><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_3"><span class="lab"><font color="Green" title="E70">A22</font></span></a>, <a class="txt" href="#E1:109_1_1_2_3_1_1_3"><span class="lab"><font color="Green" title="E73">A27</font></span></a>, <a class="txt" href="#E2:109_1_1_2_3_1_1_3"><span class="lab"><font color="Green" title="E74">A28</font></span></a>, <a class="ref" href="scmring2.html#T14" title="SCMRING2:th.14">SCMRING2:14</a></span>
<br/>.= 
<font color="Maroon" title="c4">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="txt" href="#E1:109_1_1_2_3_1_1_3"><span class="lab"><font color="Green" title="E73">A27</font></span></a>, <a class="ref" href="scmring3.html#T7" target="_self" title="SCMRING3:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_3"><span class="lab"><font color="Green" title="E70">A22</font></span></a>, <a class="txt" href="#E1:109_1_1_2_3_1_1_3"><span class="lab"><font color="Green" title="E73">A27</font></span></a>, <a class="txt" href="#E2:109_1_1_2_3_1_1_3"><span class="lab"><font color="Green" title="E74">A28</font></span></a>, <a class="ref" href="scmring2.html#T14" title="SCMRING2:th.14">SCMRING2:14</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:109_1_1_2_3"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:109_1_1"><span class="lab"><font color="Green" title="E69">A14</font></span></a>, <a class="ref" href="funct_1.html#T165" title="FUNCT_1:th.165">FUNCT_1:165</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_7.html#D2" title="FUNCT_7:def.2">FUNCT_7:def 2</a> <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="109_1_1_2_4"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_4"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K6" title="SCMRING2:func.6">SubFrom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>
 ; <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> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font>, <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E3:109_1_1_2_4"/><span class="lab"><font color="Green" title="E70">A29</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K6" title="SCMRING2:func.6">SubFrom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>
 ;<br/><a NAME="E4:109_1_1_2_4"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  holds <br/><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1_2_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  implies <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:109_1_1_2_4_1"/><span class="lab"><font color="Green" title="E71">A30</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/>

<a NAME="E2:109_1_1_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A31</font></span>: 
not <font color="Maroon" title="c7">x</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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_2_4_1_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="E1:109_1_1_2_4_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">R</font></span>)</span> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> or ( <font color="Maroon" title="c7">x</font> is    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109_1_1"><span class="lab"><font color="Green" title="E68">A13</font></span></a>, <a class="txt" href="#E1:109_1_1_2_4_1"><span class="lab"><font color="Green" title="E71">A30</font></span></a>, <a class="txt" href="#E2:109_1_1_2_4_1"><span class="lab"><font color="Green" title="E72">A31</font></span></a>, <a class="ref" href="scmring3.html#T5" target="_self" title="SCMRING3:th.5">Th5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_2_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_4_1_1_1"/><span class="lab"><font color="Green" title="E73">A32</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">R</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c3">s1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_4"><span class="lab"><font color="Green" title="E70">A29</font></span></a>, <a class="ref" href="scmring2.html#T15" title="SCMRING2:th.15">SCMRING2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1"><span class="lab"><font color="Green" title="E67">A12</font></span></a>, <a class="txt" href="#E3:109_1_1_2_4"><span class="lab"><font color="Green" title="E70">A29</font></span></a>, <a class="txt" href="#E1:109_1_1_2_4_1_1_1"><span class="lab"><font color="Green" title="E73">A32</font></span></a>, <a class="ref" href="scmring2.html#T15" title="SCMRING2:th.15">SCMRING2:15</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="109_1_1_2_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_4_1_1_2"/><span class="lab"><font color="Green" title="E73">A33</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_4"><span class="lab"><font color="Green" title="E70">A29</font></span></a>, <a class="ref" href="scmring2.html#T15" title="SCMRING2:th.15">SCMRING2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s2</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="ref" href="scmring3.html#T7" target="_self" title="SCMRING3:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s2</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s2</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="ref" href="scmring3.html#T7" target="_self" title="SCMRING3:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_4"><span class="lab"><font color="Green" title="E70">A29</font></span></a>, <a class="txt" href="#E1:109_1_1_2_4_1_1_2"><span class="lab"><font color="Green" title="E73">A33</font></span></a>, <a class="ref" href="scmring2.html#T15" title="SCMRING2:th.15">SCMRING2:15</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="109_1_1_2_4_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:109_1_1_2_4_1_1_3"/><span class="lab"><font color="Green" title="E73">A34</font></span>: 
<font color="Maroon" title="c7">x</font> is    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font>
 <span class="kw">and </span><br/><a NAME="E2:109_1_1_2_4_1_1_3"/><span class="lab"><font color="Green" title="E74">A35</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><div class="add"><span class="kw">thus </span><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_4"><span class="lab"><font color="Green" title="E70">A29</font></span></a>, <a class="txt" href="#E1:109_1_1_2_4_1_1_3"><span class="lab"><font color="Green" title="E73">A34</font></span></a>, <a class="txt" href="#E2:109_1_1_2_4_1_1_3"><span class="lab"><font color="Green" title="E74">A35</font></span></a>, <a class="ref" href="scmring2.html#T15" title="SCMRING2:th.15">SCMRING2:15</a></span>
<br/>.= 
<font color="Maroon" title="c4">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="txt" href="#E1:109_1_1_2_4_1_1_3"><span class="lab"><font color="Green" title="E73">A34</font></span></a>, <a class="ref" href="scmring3.html#T7" target="_self" title="SCMRING3:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_4"><span class="lab"><font color="Green" title="E70">A29</font></span></a>, <a class="txt" href="#E1:109_1_1_2_4_1_1_3"><span class="lab"><font color="Green" title="E73">A34</font></span></a>, <a class="txt" href="#E2:109_1_1_2_4_1_1_3"><span class="lab"><font color="Green" title="E74">A35</font></span></a>, <a class="ref" href="scmring2.html#T15" title="SCMRING2:th.15">SCMRING2:15</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:109_1_1_2_4"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:109_1_1"><span class="lab"><font color="Green" title="E69">A14</font></span></a>, <a class="ref" href="funct_1.html#T165" title="FUNCT_1:th.165">FUNCT_1:165</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_7.html#D2" title="FUNCT_7:def.2">FUNCT_7:def 2</a> <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="109_1_1_2_5"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_5"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K7" title="SCMRING2:func.7">MultBy</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>
 ; <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> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font>, <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E3:109_1_1_2_5"/><span class="lab"><font color="Green" title="E70">A36</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K7" title="SCMRING2:func.7">MultBy</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>
 ;<br/><a NAME="E4:109_1_1_2_5"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  holds <br/><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1_2_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  implies <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:109_1_1_2_5_1"/><span class="lab"><font color="Green" title="E71">A37</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/>

<a NAME="E2:109_1_1_2_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A38</font></span>: 
not <font color="Maroon" title="c7">x</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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_2_5_1_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="E1:109_1_1_2_5_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">R</font></span>)</span> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> or ( <font color="Maroon" title="c7">x</font> is    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109_1_1"><span class="lab"><font color="Green" title="E68">A13</font></span></a>, <a class="txt" href="#E1:109_1_1_2_5_1"><span class="lab"><font color="Green" title="E71">A37</font></span></a>, <a class="txt" href="#E2:109_1_1_2_5_1"><span class="lab"><font color="Green" title="E72">A38</font></span></a>, <a class="ref" href="scmring3.html#T5" target="_self" title="SCMRING3:th.5">Th5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_2_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_5_1_1_1"/><span class="lab"><font color="Green" title="E73">A39</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">R</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c3">s1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_5"><span class="lab"><font color="Green" title="E70">A36</font></span></a>, <a class="ref" href="scmring2.html#T16" title="SCMRING2:th.16">SCMRING2:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1"><span class="lab"><font color="Green" title="E67">A12</font></span></a>, <a class="txt" href="#E3:109_1_1_2_5"><span class="lab"><font color="Green" title="E70">A36</font></span></a>, <a class="txt" href="#E1:109_1_1_2_5_1_1_1"><span class="lab"><font color="Green" title="E73">A39</font></span></a>, <a class="ref" href="scmring2.html#T16" title="SCMRING2:th.16">SCMRING2:16</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="109_1_1_2_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_5_1_1_2"/><span class="lab"><font color="Green" title="E73">A40</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_5"><span class="lab"><font color="Green" title="E70">A36</font></span></a>, <a class="ref" href="scmring2.html#T16" title="SCMRING2:th.16">SCMRING2:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s2</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="ref" href="scmring3.html#T7" target="_self" title="SCMRING3:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s2</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s2</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="ref" href="scmring3.html#T7" target="_self" title="SCMRING3:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_5"><span class="lab"><font color="Green" title="E70">A36</font></span></a>, <a class="txt" href="#E1:109_1_1_2_5_1_1_2"><span class="lab"><font color="Green" title="E73">A40</font></span></a>, <a class="ref" href="scmring2.html#T16" title="SCMRING2:th.16">SCMRING2:16</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="109_1_1_2_5_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:109_1_1_2_5_1_1_3"/><span class="lab"><font color="Green" title="E73">A41</font></span>: 
<font color="Maroon" title="c7">x</font> is    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font>
 <span class="kw">and </span><br/><a NAME="E2:109_1_1_2_5_1_1_3"/><span class="lab"><font color="Green" title="E74">A42</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><div class="add"><span class="kw">thus </span><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_5"><span class="lab"><font color="Green" title="E70">A36</font></span></a>, <a class="txt" href="#E1:109_1_1_2_5_1_1_3"><span class="lab"><font color="Green" title="E73">A41</font></span></a>, <a class="txt" href="#E2:109_1_1_2_5_1_1_3"><span class="lab"><font color="Green" title="E74">A42</font></span></a>, <a class="ref" href="scmring2.html#T16" title="SCMRING2:th.16">SCMRING2:16</a></span>
<br/>.= 
<font color="Maroon" title="c4">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="txt" href="#E1:109_1_1_2_5_1_1_3"><span class="lab"><font color="Green" title="E73">A41</font></span></a>, <a class="ref" href="scmring3.html#T7" target="_self" title="SCMRING3:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_5"><span class="lab"><font color="Green" title="E70">A36</font></span></a>, <a class="txt" href="#E1:109_1_1_2_5_1_1_3"><span class="lab"><font color="Green" title="E73">A41</font></span></a>, <a class="txt" href="#E2:109_1_1_2_5_1_1_3"><span class="lab"><font color="Green" title="E74">A42</font></span></a>, <a class="ref" href="scmring2.html#T16" title="SCMRING2:th.16">SCMRING2:16</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:109_1_1_2_5"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:109_1_1"><span class="lab"><font color="Green" title="E69">A14</font></span></a>, <a class="ref" href="funct_1.html#T165" title="FUNCT_1:th.165">FUNCT_1:165</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_7.html#D2" title="FUNCT_7:def.2">FUNCT_7:def 2</a> <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="109_1_1_2_6"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_6"/>
 ex <font color="Olive" title="b1">i1</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>  st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K9" title="SCMRING2:func.9">goto</a> <font color="Olive" title="b1">i1</font>,<font color="Maroon" title="c1">R</font>
 ; <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> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font>, <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">i1</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> <span class="kw"> such that </span><br/><a NAME="E3:109_1_1_2_6"/><span class="lab"><font color="Green" title="E70">A43</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K9" title="SCMRING2:func.9">goto</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c1">R</font>
 ;<br/><a NAME="E4:109_1_1_2_6"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  holds <br/><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1_2_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  implies <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:109_1_1_2_6_1"/><span class="lab"><font color="Green" title="E71">A44</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/>

<a NAME="E2:109_1_1_2_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A45</font></span>: 
not <font color="Maroon" title="c6">x</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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_2_6_1_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="E1:109_1_1_2_6_1_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">R</font></span>)</span> or <font color="Maroon" title="c6">x</font> is    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109_1_1"><span class="lab"><font color="Green" title="E68">A13</font></span></a>, <a class="txt" href="#E1:109_1_1_2_6_1"><span class="lab"><font color="Green" title="E71">A44</font></span></a>, <a class="txt" href="#E2:109_1_1_2_6_1"><span class="lab"><font color="Green" title="E72">A45</font></span></a>, <a class="ref" href="scmring3.html#T5" target="_self" title="SCMRING3:th.5">Th5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_2_6_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_6_1_1_1"/><span class="lab"><font color="Green" title="E73">A46</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">R</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c5">i1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_6"><span class="lab"><font color="Green" title="E70">A43</font></span></a>, <a class="ref" href="scmring2.html#T17" title="SCMRING2:th.17">SCMRING2:17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_6"><span class="lab"><font color="Green" title="E70">A43</font></span></a>, <a class="txt" href="#E1:109_1_1_2_6_1_1_1"><span class="lab"><font color="Green" title="E73">A46</font></span></a>, <a class="ref" href="scmring2.html#T17" title="SCMRING2:th.17">SCMRING2:17</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="109_1_1_2_6_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_6_1_1_2"/><span class="lab"><font color="Green" title="E73">A47</font></span>: 
<font color="Maroon" title="c6">x</font> is    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_6"><span class="lab"><font color="Green" title="E70">A43</font></span></a>, <a class="ref" href="scmring2.html#T17" title="SCMRING2:th.17">SCMRING2:17</a></span>
<br/>.= 
<font color="Maroon" title="c4">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="txt" href="#E1:109_1_1_2_6_1_1_2"><span class="lab"><font color="Green" title="E73">A47</font></span></a>, <a class="ref" href="scmring3.html#T7" target="_self" title="SCMRING3:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_6"><span class="lab"><font color="Green" title="E70">A43</font></span></a>, <a class="txt" href="#E1:109_1_1_2_6_1_1_2"><span class="lab"><font color="Green" title="E73">A47</font></span></a>, <a class="ref" href="scmring2.html#T17" title="SCMRING2:th.17">SCMRING2:17</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><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:109_1_1_2_6"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:109_1_1"><span class="lab"><font color="Green" title="E69">A14</font></span></a>, <a class="ref" href="funct_1.html#T165" title="FUNCT_1:th.165">FUNCT_1:165</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_7.html#D2" title="FUNCT_7:def.2">FUNCT_7:def 2</a> <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="109_1_1_2_7"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_7"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> ex <font color="Olive" title="b2">i1</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>  st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <font color="Olive" title="b2">i1</font>
 ; <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> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font>, <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font> being    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c6">i1</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> <span class="kw"> such that </span><br/><a NAME="E3:109_1_1_2_7"/><span class="lab"><font color="Green" title="E70">A48</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <font color="Maroon" title="c6">i1</font>
 ;<br/><a NAME="E4:109_1_1_2_7"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  holds <br/><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1_2_7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  implies <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:109_1_1_2_7_1"/><span class="lab"><font color="Green" title="E71">A49</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/>

<a NAME="E2:109_1_1_2_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A50</font></span>: 
not <font color="Maroon" title="c7">x</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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E3:109_1_1_2_7_1"/><span class="lab"><font color="Green" title="E73">A51</font></span>: 
<font color="Maroon" title="c3">s1</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="ref" href="scmring3.html#T7" target="_self" title="SCMRING3:th.7">Th7</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_2_7_1_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="E1:109_1_1_2_7_1_1"/>
( ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">R</font></span>)</span> &amp; <font color="Maroon" title="c3">s1</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> ) or ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">R</font></span>)</span> &amp; <font color="Maroon" title="c3">s1</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> ) or <font color="Maroon" title="c7">x</font> is    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109_1_1"><span class="lab"><font color="Green" title="E68">A13</font></span></a>, <a class="txt" href="#E1:109_1_1_2_7_1"><span class="lab"><font color="Green" title="E71">A49</font></span></a>, <a class="txt" href="#E2:109_1_1_2_7_1"><span class="lab"><font color="Green" title="E72">A50</font></span></a>, <a class="ref" href="scmring3.html#T5" target="_self" title="SCMRING3:th.5">Th5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_2_7_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_7_1_1_1"/><span class="lab"><font color="Green" title="E74">A52</font></span>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">R</font></span>)</span> &amp; <font color="Maroon" title="c3">s1</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><div class="add"><span class="kw">thus </span><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
<font color="Maroon" title="c6">i1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_7"><span class="lab"><font color="Green" title="E70">A48</font></span></a>, <a class="txt" href="#E1:109_1_1_2_7_1_1_1"><span class="lab"><font color="Green" title="E74">A52</font></span></a>, <a class="ref" href="scmring2.html#T18" title="SCMRING2:th.18">SCMRING2:18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_7"><span class="lab"><font color="Green" title="E70">A48</font></span></a>, <a class="txt" href="#E3:109_1_1_2_7_1"><span class="lab"><font color="Green" title="E73">A51</font></span></a>, <a class="txt" href="#E1:109_1_1_2_7_1_1_1"><span class="lab"><font color="Green" title="E74">A52</font></span></a>, <a class="ref" href="scmring2.html#T18" title="SCMRING2:th.18">SCMRING2:18</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_2_7_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_7_1_1_2"/><span class="lab"><font color="Green" title="E74">A53</font></span>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">R</font></span>)</span> &amp; <font color="Maroon" title="c3">s1</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><div class="add"><span class="kw">thus </span><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c3">s1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_7"><span class="lab"><font color="Green" title="E70">A48</font></span></a>, <a class="txt" href="#E1:109_1_1_2_7_1_1_2"><span class="lab"><font color="Green" title="E74">A53</font></span></a>, <a class="ref" href="scmring2.html#T18" title="SCMRING2:th.18">SCMRING2:18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1"><span class="lab"><font color="Green" title="E67">A12</font></span></a>, <a class="txt" href="#E3:109_1_1_2_7"><span class="lab"><font color="Green" title="E70">A48</font></span></a>, <a class="txt" href="#E3:109_1_1_2_7_1"><span class="lab"><font color="Green" title="E73">A51</font></span></a>, <a class="txt" href="#E1:109_1_1_2_7_1_1_2"><span class="lab"><font color="Green" title="E74">A53</font></span></a>, <a class="ref" href="scmring2.html#T18" title="SCMRING2:th.18">SCMRING2:18</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_2_7_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_7_1_1_3"/><span class="lab"><font color="Green" title="E74">A54</font></span>: 
<font color="Maroon" title="c7">x</font> is    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_7"><span class="lab"><font color="Green" title="E70">A48</font></span></a>, <a class="ref" href="scmring2.html#T18" title="SCMRING2:th.18">SCMRING2:18</a></span>
<br/>.= 
<font color="Maroon" title="c4">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="txt" href="#E1:109_1_1_2_7_1_1_3"><span class="lab"><font color="Green" title="E74">A54</font></span></a>, <a class="ref" href="scmring3.html#T7" target="_self" title="SCMRING3:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_7"><span class="lab"><font color="Green" title="E70">A48</font></span></a>, <a class="txt" href="#E1:109_1_1_2_7_1_1_3"><span class="lab"><font color="Green" title="E74">A54</font></span></a>, <a class="ref" href="scmring2.html#T18" title="SCMRING2:th.18">SCMRING2:18</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><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:109_1_1_2_7"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:109_1_1"><span class="lab"><font color="Green" title="E69">A14</font></span></a>, <a class="ref" href="funct_1.html#T165" title="FUNCT_1:th.165">FUNCT_1:165</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_7.html#D2" title="FUNCT_7:def.2">FUNCT_7:def 2</a> <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="109_1_1_2_8"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_8"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> ex <font color="Olive" title="b2">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="scmring2.html#K8" title="SCMRING2:func.8">:=</a> <font color="Olive" title="b2">r</font>
 ; <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> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font>, <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font> being    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c6">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E3:109_1_1_2_8"/><span class="lab"><font color="Green" title="E70">A55</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="scmring2.html#K8" title="SCMRING2:func.8">:=</a> <font color="Maroon" title="c6">r</font>
 ;<br/><a NAME="E4:109_1_1_2_8"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  holds <br/><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1_2_8_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  implies <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:109_1_1_2_8_1"/><span class="lab"><font color="Green" title="E71">A56</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/>

<a NAME="E2:109_1_1_2_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A57</font></span>: 
not <font color="Maroon" title="c7">x</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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_2_8_1_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="E1:109_1_1_2_8_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">R</font></span>)</span> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> or ( <font color="Maroon" title="c7">x</font> is    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109_1_1"><span class="lab"><font color="Green" title="E68">A13</font></span></a>, <a class="txt" href="#E1:109_1_1_2_8_1"><span class="lab"><font color="Green" title="E71">A56</font></span></a>, <a class="txt" href="#E2:109_1_1_2_8_1"><span class="lab"><font color="Green" title="E72">A57</font></span></a>, <a class="ref" href="scmring3.html#T5" target="_self" title="SCMRING3:th.5">Th5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_2_8_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_8_1_1_1"/><span class="lab"><font color="Green" title="E73">A58</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">R</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c3">s1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_8"><span class="lab"><font color="Green" title="E70">A55</font></span></a>, <a class="ref" href="scmring2.html#T19" title="SCMRING2:th.19">SCMRING2:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1"><span class="lab"><font color="Green" title="E67">A12</font></span></a>, <a class="txt" href="#E3:109_1_1_2_8"><span class="lab"><font color="Green" title="E70">A55</font></span></a>, <a class="txt" href="#E1:109_1_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E73">A58</font></span></a>, <a class="ref" href="scmring2.html#T19" title="SCMRING2:th.19">SCMRING2:19</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="109_1_1_2_8_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:109_1_1_2_8_1_1_2"/><span class="lab"><font color="Green" title="E73">A59</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
<font color="Maroon" title="c6">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_8"><span class="lab"><font color="Green" title="E70">A55</font></span></a>, <a class="ref" href="scmring2.html#T19" title="SCMRING2:th.19">SCMRING2:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_8"><span class="lab"><font color="Green" title="E70">A55</font></span></a>, <a class="txt" href="#E1:109_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E73">A59</font></span></a>, <a class="ref" href="scmring2.html#T19" title="SCMRING2:th.19">SCMRING2:19</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="109_1_1_2_8_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:109_1_1_2_8_1_1_3"/><span class="lab"><font color="Green" title="E73">A60</font></span>: 
<font color="Maroon" title="c7">x</font> is    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font>
 <span class="kw">and </span><br/><a NAME="E2:109_1_1_2_8_1_1_3"/><span class="lab"><font color="Green" title="E74">A61</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><div class="add"><span class="kw">thus </span><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_8"><span class="lab"><font color="Green" title="E70">A55</font></span></a>, <a class="txt" href="#E1:109_1_1_2_8_1_1_3"><span class="lab"><font color="Green" title="E73">A60</font></span></a>, <a class="txt" href="#E2:109_1_1_2_8_1_1_3"><span class="lab"><font color="Green" title="E74">A61</font></span></a>, <a class="ref" href="scmring2.html#T19" title="SCMRING2:th.19">SCMRING2:19</a></span>
<br/>.= 
<font color="Maroon" title="c4">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="txt" href="#E1:109_1_1_2_8_1_1_3"><span class="lab"><font color="Green" title="E73">A60</font></span></a>, <a class="ref" href="scmring3.html#T7" target="_self" title="SCMRING3:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1_2_8"><span class="lab"><font color="Green" title="E70">A55</font></span></a>, <a class="txt" href="#E1:109_1_1_2_8_1_1_3"><span class="lab"><font color="Green" title="E73">A60</font></span></a>, <a class="txt" href="#E2:109_1_1_2_8_1_1_3"><span class="lab"><font color="Green" title="E74">A61</font></span></a>, <a class="ref" href="scmring2.html#T19" title="SCMRING2:th.19">SCMRING2:19</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:109_1_1_2_8"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s2</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:109_1_1"><span class="lab"><font color="Green" title="E69">A14</font></span></a>, <a class="ref" href="funct_1.html#T165" title="FUNCT_1:th.165">FUNCT_1:165</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_7.html#D2" title="FUNCT_7:def.2">FUNCT_7:def 2</a> <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>
