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<span class="kw">let </span><font color="Maroon" title="c2">s</font> be   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a><br/>  for <font color="Olive" title="b2">I</font> being   <a href="compos_1.html#NM5" title="COMPOS_1:NM.5">Program</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a><br/>  for <font color="Olive" title="b3">a</font> being    <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a>   st  not <font color="Olive" title="b2">I</font> <a href="scmfsa7b.html#R4" title="SCMFSA7B:pred.4">destroys</a> <font color="Olive" title="b3">a</font> &amp; <font color="Olive" title="b2">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c2">s</font>,<font color="Olive" title="b1">P</font> holds <br/> for <font color="Olive" title="b4">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Olive" title="b1">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b2">I</font></span>)</span>,<span class="p2">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c2">s</font></span>)</span>,<font color="Olive" title="b4">k</font>)</span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Olive" title="b3">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font> be   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">I</font> being   <a href="compos_1.html#NM5" title="COMPOS_1:NM.5">Program</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a><br/>  for <font color="Olive" title="b2">a</font> being    <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a>   st  not <font color="Olive" title="b1">I</font> <a href="scmfsa7b.html#R4" title="SCMFSA7B:pred.4">destroys</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">P</font> holds <br/> for <font color="Olive" title="b3">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b1">I</font></span>)</span>,<span class="p2">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c2">s</font></span>)</span>,<font color="Olive" title="b3">k</font>)</span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Olive" title="b2">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">I</font> be   <a href="compos_1.html#NM5" title="COMPOS_1:NM.5">Program</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being    <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a>   st  not <font color="Maroon" title="c4">I</font> <a href="scmfsa7b.html#R4" title="SCMFSA7B:pred.4">destroys</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c4">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">P</font> holds <br/> for <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p2">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c2">s</font></span>)</span>,<font color="Olive" title="b2">k</font>)</span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Olive" title="b1">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">a</font> be    <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">I</font> <a href="scmfsa7b.html#R4" title="SCMFSA7B:pred.4">destroys</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">P</font> implies  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p2">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c2">s</font></span>)</span>,<font color="Olive" title="b1">k</font>)</span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:36"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
 not <font color="Maroon" title="c4">I</font> <a href="scmfsa7b.html#R4" title="SCMFSA7B:pred.4">destroys</a> <font color="Maroon" title="c5">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">P</font> or  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p2">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c2">s</font></span>)</span>,<font color="Olive" title="b1">k</font>)</span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font> )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p2">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c2">s</font></span>)</span>,$1)</span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font>;<br/>
<a NAME="E2:36"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<font color="Maroon" title="c4">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T25" title="FUNCT_4:th.25">FUNCT_4:25</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:36"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c4">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p2">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c2">s</font></span>)</span>,<font color="Olive" title="b1">k</font>)</span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font></span><br/>

<div><span class="lab"><font color="Green" title="E21">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:36_1"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><span class="kw">set </span><font color="Maroon" title="c7">l</font> =  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p2">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c6">k</font>)</span>)</span>;<br/><a NAME="E2:36_1"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
 <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p2">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="scmfsa7b.html#D6" target="_self" title="SCMFSA7B:def.6">Def7</a></span>;<br/><a NAME="E3:36_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="compos_1.html#K3" title="COMPOS_1:func.3">.</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p2">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p3">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c6">k</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p2">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p3">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c6">k</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>;<br/><a NAME="E4:36_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="compos_1.html#K3" title="COMPOS_1:func.3">.</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p2">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p3">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c6">k</font>)</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E5:36_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A8</font></span>: 
 not <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="compos_1.html#K3" title="COMPOS_1:func.3">.</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p2">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p3">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c6">k</font>)</span>)</span></span>)</span> <a href="scmfsa7b.html#R3" title="SCMFSA7B:pred.3">destroys</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="scmfsa7b.html#D4" target="_self" title="SCMFSA7B:def.4">Def4</a></span>;<br/><a NAME="E6:36_1"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p2">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c2">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font> = 
<span class="p1">(<span class="default"><a href="extpro_1.html#K4" title="EXTPRO_1:func.4">Following</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p2">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p3">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c6">k</font>)</span>)</span>)</span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="extpro_1.html#T3" title="EXTPRO_1:th.3">EXTPRO_1:3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="extpro_1.html#K2" title="EXTPRO_1:func.2">Exec</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="compos_1.html#K3" title="COMPOS_1:func.3">.</a> <span class="p3">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p4">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p5">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c6">k</font>)</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p3">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c6">k</font>)</span>)</span>)</span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>)</span>)</span></span>)</span>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="ref" href="scmfsa7b.html#T20" target="_self" title="SCMFSA7B:th.20">Th26</a></span>
<br/>.= 
<font color="Maroon" title="c2">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E8:36_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E5:36"/><span class="lab"><font color="Green" title="E22">A10</font></span>: 
 not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa_2.html#T102" title="SCMFSA_2:th.102">SCMFSA_2:102</a></span>;<br/>
<span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p2">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c2">s</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font> = 
<span class="p1">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="extpro_1.html#T2" title="EXTPRO_1:th.2">EXTPRO_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c2">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="ref" href="funct_4.html#T11" title="FUNCT_4:th.11">FUNCT_4:11</a></span>
;<br/>
<a NAME="E7:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A11</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<span class="kw">thus </span><a NAME="E8:36"/>
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E7:36"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="txt" href="#E4:36"><span class="lab"><font color="Green" title="E21">A5</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
