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<b>let </b><font color="Maroon">s</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:25"/><i><font color="Green">E37</font></i>: 
 <a href="ami_1.html#K6">IC</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K5">insloc</a> 0
 <b>and </b><br/><a NAME="E2:25"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="scmfsa_2.html#M1">Int-Location</a> ;<br/><b>let </b><font color="Maroon">k</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E3:25"/><i><font color="Green">E43</font></i>: 
 <a href="scmfsa_7.html#K1" target="_self">Load</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <font color="Maroon">a</font>,<font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E4:25"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="scmfsa_2.html#K4">intloc</a> 0
 ;<br/>

<a NAME="E5:25"/><i><font color="Green">E45</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K1" target="_self">Load</a> <span class="p2">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <font color="Maroon">a</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <font color="Maroon">a</font>,<font color="Maroon">k</font></span>)</span> </span>}</span>  &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K1" target="_self">Load</a> <span class="p2">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <font color="Maroon">a</font>,<font color="Maroon">k</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="scmfsa_7.html#K1" target="_self">Load</a> <span class="p2">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <font color="Maroon">a</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <font color="Maroon">a</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:25_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <font color="Maroon">a</font>,<font color="Maroon">k</font></span>)</span>
 ;<br/><a NAME="E2:25_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <font color="Maroon">a</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:25_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:25_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i/>;<br/><a NAME="E5:25_1"/><i><font color="Green">E51</font></i>: 
 <a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K1" target="_self">Load</a> <span class="p2">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <font color="Maroon">a</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="scmfsa_7.html#K1" target="_self">Load</a> <span class="p2">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <font color="Maroon">a</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <font color="Maroon">a</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <font color="Maroon">a</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E7:25_1"/>
<span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <font color="Maroon">a</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> <font color="Maroon">c</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <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>hereby </b>
<div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:25_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <font color="Maroon">a</font>,<font color="Maroon">k</font></span>)</span>
 ;<br/><a NAME="E2:25_2"/><b>then </b>
(  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> &amp; ( for <font color="Olive">b</font> being   <a href="scmfsa_2.html#M1">Int-Location</a>   st <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Olive">b</font> ) &amp; ( for <font color="Olive">f</font> being   <a href="scmfsa_2.html#M2">FinSeq-Location</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K21">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmfsa_2.html#K21">.</a> <font color="Olive">f</font> ) )
 <b>by </b><i>, , <a class="ref" href="scmfsa_7.html#T1" target="_self">Th1</a>, <a class="txt" href="scmfsa_7.html#E2"><i><font color="Green">Lemma26</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:25_2"/>
(  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">i</font> &amp; ( for <font color="Olive">b</font> being   <a href="scmfsa_2.html#M1">Int-Location</a>   st <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Olive">b</font> ) &amp; ( for <font color="Olive">f</font> being   <a href="scmfsa_2.html#M2">FinSeq-Location</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K21">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmfsa_2.html#K21">.</a> <font color="Olive">f</font> ) )
 ;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E8:25"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <font color="Maroon">a</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="scmfsa_7.html#T1" target="_self">Th1</a>, <a class="txt" href="scmfsa_7.html#E2"><i><font color="Green">Lemma26</font></i></a>, </i>;<br/>


</div>
