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<span class="kw">let </span><font color="Maroon" title="c1">k</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">R</font> being  <a href="scmring1.html#V1" title="SCMRING1:attr.1">good</a>  <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a><br/> for <font color="Olive" title="b2">s1</font>, <font color="Olive" title="b3">s2</font> being  <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="Olive" title="b1">R</font></span>)</span><br/> for <font color="Olive" title="b4">p</font> being  <a href="ami_1.html#V10" title="AMI_1:attr.10">autonomic</a>  <a href="ami_1.html#NM4" title="AMI_1:NM.4">FinPartState</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Olive" title="b1">R</font></span>)</span>  st <font color="Olive" title="b4">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">s1</font> &amp;  <a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Olive" title="b4">p</font>,<font color="Maroon" title="c1">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">s2</font> holds <br/><font color="Olive" title="b4">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">s1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <font color="Olive" title="b3">s2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">R</font> be   <a href="scmring1.html#V1" title="SCMRING1:attr.1">good</a>  <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">s1</font>, <font color="Olive" title="b2">s2</font> being  <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="c2">R</font></span>)</span><br/> for <font color="Olive" title="b3">p</font> being  <a href="ami_1.html#V10" title="AMI_1:attr.10">autonomic</a>  <a href="ami_1.html#NM4" title="AMI_1:NM.4">FinPartState</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span>  st <font color="Olive" title="b3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">s1</font> &amp;  <a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Olive" title="b3">p</font>,<font color="Maroon" title="c1">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">s2</font> holds <br/><font color="Olive" title="b3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">s1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <font color="Olive" title="b2">s2</font></span>)</span></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="c2">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"> for <font color="Olive" title="b1">p</font> being  <a href="ami_1.html#V10" title="AMI_1:attr.10">autonomic</a>  <a href="ami_1.html#NM4" title="AMI_1:NM.4">FinPartState</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span>  st <font color="Olive" title="b1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">s1</font> &amp;  <a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">s2</font> holds <br/><font color="Olive" title="b1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <font color="Maroon" title="c4">s2</font></span>)</span></span><br/>







<a NAME="E1:35"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
 <a href="ami_1.html#K15" title="AMI_1:func.15">Data-Locations</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmring2.html#T31" title="SCMRING2:th.31">SCMRING2:31</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">s3</font> =  <a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <font color="Maroon" title="c4">s2</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">s</font> = <font color="Maroon" title="c3">s1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <font color="Maroon" title="c4">s2</font></span>)</span> as   <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="c2">R</font></span>)</span> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">p</font> be   <a href="ami_1.html#V10" title="AMI_1:attr.10">autonomic</a>  <a href="ami_1.html#NM4" title="AMI_1:NM.4">FinPartState</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">s1</font> &amp;  <a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c1">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">s2</font> implies <font color="Maroon" title="c7">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <font color="Maroon" title="c4">s2</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:35"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">s1</font>
 <span class="kw">and </span><br/><a NAME="E4:35"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
 <a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c1">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">s2</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="c7">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <font color="Maroon" title="c4">s2</font></span>)</span></span><br/>

<a NAME="E5:35"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">R</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/>
<a NAME="E6:35"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A5</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p4">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="scmring4.html#T1" target="_self" title="SCMRING4:th.1">Th1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E27">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><span class="kw">now </span></a><div class="add"> <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">s2</font> = 
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="c2">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 href="ami_2.html#K3" title="AMI_2:func.3">SCM-Memory</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmring2.html#D1" title="SCMRING2:def.1">SCMRING2:def 1</a></span>
;<br/><a NAME="E2:35_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A7</font></span>: 
 <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c8">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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">p</font> implies <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E3:35_1"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">p</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="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><a NAME="E4:35_1"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</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="c2">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a>  or <font color="Maroon" title="c8">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="txt" href="#E6:35"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_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="E1:35_1_2"/>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</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="c2">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a>  or <font color="Maroon" title="c8">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_2_1"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</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="c2">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:35_1_2_1"/>
not  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:35_1_2_1_1"/>
 <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:35_1_2_1_1"/><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="c2">R</font></span>)</span> is    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c2">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmring2.html#T1" title="SCMRING2:th.1">SCMRING2:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:35_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmring3.html#T3" title="SCMRING3:th.3">SCMRING3:3</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><a NAME="E3:35_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A11</font></span>: 
not  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <font color="Maroon" title="c4">s2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:35_1"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/><a NAME="E4:35_1_2_1"/>
<font color="Maroon" title="c8">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="c2">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_2_1"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E5:35_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_2_1"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_2_1"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</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="35_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_2_2"/><span class="lab"><font color="Green" title="E31">A12</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c9">DPRp</font> =  <a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c1">k</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c10">DPp</font> =  <a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <font color="Maroon" title="c7">p</font>;<br/><a NAME="E2:35_1_2_2"/>
<font color="Maroon" title="c8">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> <font color="Maroon" title="c7">p</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E1:35_1_2_2"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:35_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A13</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <font color="Maroon" title="c7">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/><a NAME="E4:35_1_2_2"/>
 <a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="amistd_2.html#T68" title="AMISTD_2:th.68">AMISTD_2:68</a></span>;<br/><a NAME="E5:35_1_2_2"/><span class="kw">then </span>
 <a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <font color="Maroon" title="c7">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="amistd_2.html#K13" title="AMISTD_2:func.13">Relocated</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T88" title="RELAT_1:th.88">RELAT_1:88</a></span>;<br/><a NAME="E6:35_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A14</font></span>: 
 <a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <font color="Maroon" title="c7">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E7:35_1_2_2"/><span class="kw">then </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#K24" title="AMI_1:func.24">DataPart</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></span>;<br/><a NAME="E8:35_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">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> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_2_2"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E3:35_1_2_2"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E9:35_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A15</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <font color="Maroon" title="c4">s2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/><a NAME="E10:35_1_2_2"/><span class="lab"><font color="Green" title="E35">A16</font></span>: 
<font color="Maroon" title="c4">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <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="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:35_1_2_2"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/><a NAME="E11:35_1_2_2"/>
 <a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <font color="Maroon" title="c7">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T88" title="RELAT_1:th.88">RELAT_1:88</a></span>;<br/><a NAME="E12:35_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A17</font></span>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_2_2"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></span>;<br/><a NAME="E13:35_1_2_2"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_2_2"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E6:35_1_2_2"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E14:35_1_2_2"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:35_1_2_2"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E10:35_1_2_2"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E9:35_1_2_2"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2_3"><span class="kw">suppose </span></a><a NAME="E1:35_1_2_3"/><span class="lab"><font color="Green" title="E31">A18</font></span>: 
<font color="Maroon" title="c8">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> 
 ; <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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_2_3_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <font color="Maroon" title="c4">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"> contradiction</span><br/><a NAME="E2:35_1_2_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">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> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/><a NAME="E3:35_1_2_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_2_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_2_3"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="ref" href="ami_2.html#T29" title="AMI_2:th.29">AMI_2:29</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</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><a NAME="E3:35_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_2_3"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</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>
<a NAME="E8:35"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:35"/>
<font color="Maroon" title="c7">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K24" title="AMI_1:func.24">DataPart</a> <font color="Maroon" title="c4">s2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</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/>


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