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

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p</font> be  <a href="ami_1.html#V11">autonomic</a>  <a href="ami_1.html#M1">FinPartState</a> of  <a href="ami_3.html#K1">SCM</a> ;<br/><b>let </b><font color="Maroon">s1</font> be   <a href="ami_1.html#NM4">State</a> of <a href="ami_3.html#K1">SCM</a> , <font color="Maroon">s2</font> be   <a href="ami_1.html#NM4">State</a> of <a href="ami_3.html#K1">SCM</a> ;<br/>





<b>assume </b><a NAME="E1:34"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">s1</font> &amp;  <a href="reloc.html#K6" target="_self">Relocated</a> <font color="Maroon">p</font>,<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">s2</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s1</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="relat_1.html#K7">|</a> <a href="ami_2.html#K2">SCM-Data-Loc</a> </span>)</span> as   <a href="ami_1.html#NM4">State</a> of <a href="ami_3.html#K1">SCM</a>  <b>by </b><i><a class="ref" href="ami_5.html#T82">AMI_5:82</a></i>;<br/>
<b>set </b><font color="Maroon">s3</font> = <font color="Maroon">s2</font> <a href="relat_1.html#K7">|</a> <a href="ami_2.html#K2">SCM-Data-Loc</a> ;<br/>
<a NAME="E3:34"/><i><font color="Green">E27</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="ami_3.html#K1">SCM</a> </span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <a href="ami_2.html#K2">SCM-Data-Loc</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="ami_2.html#K3">SCM-Instr-Loc</a> 
 
<b>by </b><i><a class="ref" href="ami_3.html#T37">AMI_3:37</a>, <a class="ref" href="ami_5.html#T23">AMI_5:23</a></i>;<br/>
<a NAME="E4:34"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="ami_3.html#T36">AMI_3:36</a>, <a class="ref" href="ami_5.html#T23">AMI_5:23</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:34_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:34_1"/>
 <a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s2</font>
 <b>by </b><i><a class="ref" href="ami_5.html#T27">AMI_5:27</a></i>;<br/><a NAME="E3:34_1"/><b>then </b>
 <a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">s2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="ami_2.html#K2">SCM-Data-Loc</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E4:34_1"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="relat_1.html#K7">|</a> <a href="ami_2.html#K2">SCM-Data-Loc</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_2.html#K2">SCM-Data-Loc</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E5:34_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="ami_3.html#K1">SCM</a> </span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <a href="ami_2.html#K2">SCM-Data-Loc</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K3">SCM-Instr-Loc</a>  )
 <b>by </b><i><a class="ref" href="reloc.html#D3" target="_self">Def3</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="ami_3.html#K1">SCM</a> </span>)</span></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K2">SCM-Data-Loc</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K3">SCM-Instr-Loc</a>  )
 </span><b>by </b><i><a class="ref" href="reloc.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="ami_3.html#K1">SCM</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:34_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K2">IC</a> <a href="ami_3.html#K1">SCM</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:34_1_1_1"/>
not  <a href="ami_1.html#K2">IC</a> <a href="ami_3.html#K1">SCM</a>  <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K2">SCM-Data-Loc</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1_1_1_1"/>
 <a href="ami_1.html#K2">IC</a> <a href="ami_3.html#K1">SCM</a>  <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K2">SCM-Data-Loc</a> 
 ;<br/>

<a NAME="E2:34_1_1_1_1"/><b>then </b>
 <a href="ami_1.html#K2">IC</a> <a href="ami_3.html#K1">SCM</a>  is    <a href="ami_3.html#M1">Data-Location</a> 
 
<b>by </b><i><a class="ref" href="ami_3.html#D2">AMI_3:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:34_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="ami_5.html#T20">AMI_5:20</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:34_1_1_1"/><b>then </b>
<font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="reloc.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E5:34_1_1_1"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K2">SCM-Data-Loc</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">DPp</font> =  <a href="ami_5.html#K7">DataPart</a> <font color="Maroon">p</font>;<br/><a NAME="E2:34_1_1_2"/><i><font color="Green">E65</font></i>: 
 <a href="ami_5.html#K7">DataPart</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <a href="ami_2.html#K2">SCM-Data-Loc</a> 
 <b>by </b><i><a class="ref" href="ami_5.html#T96">AMI_5:96</a></i>;<br/><a NAME="E3:34_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="ami_2.html#K2">SCM-Data-Loc</a> 
 <b>by </b><i>, <a class="ref" href="reloc.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:34_1_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ami_5.html#K7">DataPart</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="reloc.html#T8" target="_self">Th8</a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E5:34_1_1_2"/>
 <a href="ami_5.html#K7">DataPart</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T88">RELAT_1:88</a></i>;<br/><a NAME="E6:34_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><a href="ami_5.html#K7">DataPart</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="reloc.html#T9" target="_self">Th9</a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/><a NAME="E7:34_1_1_2"/>
 <a href="ami_5.html#K7">DataPart</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="ami_5.html#K7">DataPart</a> <span class="p1">(<span class="default"><a href="reloc.html#K6" target="_self">Relocated</a> <font color="Maroon">p</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="reloc.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E8:34_1_1_2"/><b>then </b>
 <a href="ami_5.html#K7">DataPart</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="reloc.html#K6" target="_self">Relocated</a> <font color="Maroon">p</font>,<font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T88">RELAT_1:88</a></i>;<br/><a NAME="E9:34_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="ami_5.html#K7">DataPart</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">s2</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E10:34_1_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="ami_5.html#K7">DataPart</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="reloc.html#T9" target="_self">Th9</a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/><a NAME="E11:34_1_1_2"/><i><font color="Green">E72</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ami_5.html#K7">DataPart</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s2</font>
 <b>by </b><i><a class="ref" href="reloc.html#T11" target="_self">Th11</a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/><a NAME="E12:34_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="relat_1.html#K7">|</a> <a href="ami_2.html#K2">SCM-Data-Loc</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="reloc.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E13:34_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">s2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="ami_2.html#K2">SCM-Data-Loc</a> 
 <b>by </b><i><a class="ref" href="reloc.html#T7" target="_self">Th7</a>, <a class="ref" href="reloc.html#T9" target="_self">Th9</a>, <a class="ref" href="reloc.html#T13" target="_self">Th13</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E14:34_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="relat_1.html#K7">|</a> <a href="ami_2.html#K2">SCM-Data-Loc</a> </span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><b>hence </b><a NAME="E15:34_1_1_2"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="reloc.html#T10" target="_self">Th10</a>, <a class="ref" href="reloc.html#T12" target="_self">Th12</a>, <a class="txt" href="reloc.html#E4:34"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_3"><b>suppose </b></a><a NAME="E1:34_1_1_3"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K3">SCM-Instr-Loc</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_1_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="relat_1.html#K7">|</a> <a href="ami_2.html#K2">SCM-Data-Loc</a> </span>)</span>
 ;<br/><a NAME="E2:34_1_1_3_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">s2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="ami_2.html#K2">SCM-Data-Loc</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E3:34_1_1_3_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K2">SCM-Data-Loc</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:34_1_1_3_1"/>
contradiction
 <b>by </b><i><a class="ref" href="reloc.html#T7" target="_self">Th7</a>, <a class="ref" href="ami_5.html#T33">AMI_5:33</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><a NAME="E3:34_1_1_3"/><b>then </b>
<font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E4:34_1_1_3"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:34"/>
<font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">s1</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="relat_1.html#K7">|</a> <a href="ami_2.html#K2">SCM-Data-Loc</a> </span>)</span>
 <b>by </b><i><a class="ref" href="reloc.html#T5" target="_self">Th5</a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>


</div>
