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<b>let </b><font color="Maroon">R</font> be  <a href="scmring1.html#V1">good</a> <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">s1</font> be   <a href="ami_1.html#NM4">State</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>;<br/><b>let </b><font color="Maroon">s2</font> be   <a href="ami_1.html#NM4">State</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:42"/><i><font color="Green">E45</font></i>: 
 <a href="ami_1.html#K6">IC</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K6">IC</a> <font color="Maroon">s2</font>
 <b>and </b><br/><a NAME="E2:42"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">a</font> being   <a href="scmring2.html#M1">Data-Location</a> of <font color="Maroon">R</font> holds  <font color="Maroon">s1</font> <a href="scmring2.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="scmring2.html#K3">.</a> <font color="Olive">a</font>
 <b>and </b><br/><a NAME="E3:42"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">i</font> being  <a href="ami_1.html#NM2">Instruction-Location</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span> holds  <font color="Maroon">s1</font> <a href="ami_1.html#K13">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="ami_1.html#K13">.</a> <font color="Olive">i</font>
 ;<br/>

<a NAME="E4:42"/>
<font color="Maroon">s1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="scmring1.html#K2">SCM-OK</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="scmring2.html#D1">SCMRING2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">g1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:42"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmring1.html#K2">SCM-OK</a> <font color="Maroon">R</font></span>)</span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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="scmring1.html#K2">SCM-OK</a> <font color="Maroon">R</font></span>)</span> holds <br/><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="scmring1.html#K2">SCM-OK</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E7:42"/>
<font color="Maroon">s2</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="scmring1.html#K2">SCM-OK</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="scmring2.html#D1">SCMRING2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">g2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:42"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmring1.html#K2">SCM-OK</a> <font color="Maroon">R</font></span>)</span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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="scmring1.html#K2">SCM-OK</a> <font color="Maroon">R</font></span>)</span> holds <br/><font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="scmring1.html#K2">SCM-OK</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E10:42"/><i><font color="Green">E74</font></i>: 
(  <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font> &amp;  <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_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="E1:42_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/> <a href="numbers.html#K5">NAT</a>  = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="scmring2.html#D1">SCMRING2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p4">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</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/>
;<br/><a NAME="E3:42_1"/><b>then </b><i><font color="Green">E77</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</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="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</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="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1"><b>suppose </b></a><a NAME="E1:42_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:42_1_2_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:42_1_2_1"/>
<font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p2">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <span class="p2">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:42_1_2_1"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="txt" href="scmring4.html#E1"><i><font color="Green">Lemma38</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_2"><b>suppose </b></a><a NAME="E1:42_1_2_2"/>
<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"><a NAME="E2:42_1_2_2"/><b>then </b>
<font color="Maroon">x</font> is    <a href="scmring2.html#M1">Data-Location</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="scmring2.html#T1">SCMRING2:1</a></i>;<br/><b>hence </b><a NAME="E3:42_1_2_2"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_3"><b>suppose </b></a><a NAME="E1:42_1_2_3"/><i><font color="Green">E79</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"><a NAME="E2:42_1_2_3"/>
the <a href="ami_1.html#U2">Instruction-Locations</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_2.html#K3">SCM-Instr-Loc</a> 
 <b>by </b><i><a class="ref" href="scmring2.html#D1">SCMRING2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:42_1_2_3"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , <a class="txt" href="scmring4.html#E1"><i><font color="Green">Lemma38</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E12:42"/>
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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