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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of the <a href="ami_1.html#U4">Instructions</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  =  <a href="scmpds_2.html#K4">goto</a> 0;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">a<sub>1</sub></font>,0</span>)</span>;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> <b> such that </b><br/><a NAME="E2:2_1_1"/><i><font color="Green">E32</font></i>: 
(  <a href="ami_1.html#K6">IC</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> 0 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">s</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>) &amp; <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Olive">i</font>,0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">i</font>) ) ) )
 <b>from </b><i><a class="ref" href="scmpds_4.html#S1">SCMPDS_4:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">s</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_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>thus </b><a NAME="E1:2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K2">SCM-Data-Loc</a>  implies <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_1_1_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/>

<a NAME="E2:2_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> is    <a href="scmpds_2.html#M1">Int_position</a> 
 
<b>by </b><i><a class="ref" href="scmpds_2.html#T9">SCMPDS_2:9</a></i>;<br/>
<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:2_1_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/>
<font color="Maroon">x</font> = 
 <a href="scmp_gcd.html#K1">intpos</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">i</font>,0
<b>by </b><i><a class="ref" href="scmp_gcd.html#T5">SCMP_GCD:5</a></i>
;<br/>
<b>hence </b><a NAME="E6:2_1_1_1_1"/>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:2_1_1_1"/>
( <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>  implies <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K4">goto</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_1_1_1_2"/>
<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> 
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:2_1_1_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="scmpds_3.html#T32">SCMPDS_3:32</a></i>;<br/>
<b>thus </b><a NAME="E4:2_1_1_1_2"/>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K4">goto</a> 0
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:2_1_1_1"/>
( <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>  implies <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> 0 )
 <b>by </b><i>, <a class="ref" href="ami_1.html#D15">AMI_1:def 15</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:2_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K2">SCM-Data-Loc</a>  implies <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) &amp; ( <font color="Olive">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>  implies <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K4">goto</a> 0 ) &amp; ( <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a>  implies <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> 0 ) )
 ;<br/>


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