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<div class="add">

<b>let </b><font color="Maroon">s1</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/><b>let </b><font color="Maroon">s2</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/>



<b>set </b><font color="Maroon">X</font> = <a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="xboole_0.html#K2">\/</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="scmpds_2.html#K1">SCMPDS</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>;<br/>
<b>set </b><font color="Maroon">Y</font> = the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/>
<a NAME="E1:66"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="scmpds_2.html#K1">SCMPDS</a> 
 
<b>by </b><i><a class="ref" href="ami_3.html#T36">AMI_3:36</a></i>;<br/>
<a NAME="E2:66"/><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="scmpds_2.html#K1">SCMPDS</a> 
 
<b>by </b><i><a class="ref" href="ami_3.html#T36">AMI_3:36</a></i>;<br/>
<i><font color="Green">E40</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="xboole_0.html#K2">\/</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span> <a href="subset_1.html#K6">\</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="xboole_0.html#K2">\/</a> <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="scmpds_2.html#K1">SCMPDS</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default">the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  <a href="subset_1.html#K4">\/</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="xboole_0.html#K2">\/</a> <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="scmpds_2.html#K1">SCMPDS</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>

;<br/>
<a NAME="E4:66"/><i><font color="Green">E45</font></i>: 
the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span> <a href="subset_1.html#K6">\</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E5:66"/><i><font color="Green">E76</font></i>: 
<a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="xboole_0.html#K2">\/</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="scmpds_2.html#K1">SCMPDS</a> </span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_2"/>
<a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="xboole_0.html#K2">\/</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="scmpds_2.html#K1">SCMPDS</a> </span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:66_2"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="xboole_0.html#K2">\/</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="scmpds_2.html#K1">SCMPDS</a> </span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:66_2"/><i><font color="Green">E78</font></i>: 
( <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> <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="scmpds_2.html#K1">SCMPDS</a> </span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="scmpds_5.html#T7" target="_self">Th7</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="66_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1"/>
( <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#R1">=</a>  <a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a>  )
 </span><b>by </b><i><a class="ref" href="scmpds_5.html#T8" target="_self">Th8</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1"><b>suppose </b></a><a NAME="E1:66_2_1_1"/>
<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>hence </b><a NAME="E2:66_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="scmpds_5.html#T7" target="_self">Th7</a>, <a class="ref" href="scmpds_2.html#T10">SCMPDS_2:10</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2"><b>suppose </b></a><a NAME="E1:66_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_2_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="scmpds_5.html#T7" target="_self">Th7</a>, <a class="ref" href="ami_1.html#T48">AMI_1:48</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:66"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">s1</font></span>)</span> <a href="subset_1.html#K6">\</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  <a href="hidden.html#R1">=</a> <a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="xboole_0.html#K2">\/</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="scmpds_2.html#K1">SCMPDS</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="scmpds_5.html#T4" target="_self">Th4</a>, <a class="ref" href="scmpds_5.html#T5" target="_self">Th5</a>, <a class="ref" href="scmpds_3.html#T5">SCMPDS_3:5</a>, <a class="ref" href="xboole_1.html#T72">XBOOLE_1:72</a></i>;<br/>
<a NAME="E7:66"/>
<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#K6">\</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  <a href="hidden.html#R1">=</a> <a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="xboole_0.html#K2">\/</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="scmpds_2.html#K1">SCMPDS</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="scmpds_5.html#E4:66"><i><font color="Green">E45</font></i></a>, <a class="ref" href="scmpds_5.html#T4" target="_self">Th4</a>, <a class="ref" href="scmpds_5.html#T5" target="_self">Th5</a>, <a class="ref" href="scmpds_5.html#T6" target="_self">Th6</a>, <a class="ref" href="scmpds_3.html#T5">SCMPDS_3:5</a>, <a class="ref" href="xboole_1.html#T72">XBOOLE_1:72</a></i>;<br/>
<b>hence </b><a NAME="E8:66"/>
( <font color="Maroon">s1</font>,<font color="Maroon">s2</font> <a href="funct_7.html#R1">equal_outside</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  iff <font color="Maroon">s1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="xboole_0.html#K2">\/</a> <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="scmpds_2.html#K1">SCMPDS</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="xboole_0.html#K2">\/</a> <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="scmpds_2.html#K1">SCMPDS</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> )
 <b>by </b><i><a class="ref" href="scmpds_5.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_7.html#D2">FUNCT_7:def 2</a></i>;<br/>


</div>
