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<div class="add">

<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="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</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">s</font> being   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</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="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  st (  for <font color="Olive" title="b3">t</font> being   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="memstr_0.html#V4" title="MEMSTR_0:attr.4">-started</a>  <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a><br/>  for <font color="Olive" title="b4">Q</font> being   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  st  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Olive" title="b3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Olive" title="b1">s</font> holds <br/><font color="Olive" title="b2">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b3">t</font>,<font color="Olive" title="b4">Q</font> ) holds <br/><font color="Olive" title="b2">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c3">P</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">s</font> be   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</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="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  st (  for <font color="Olive" title="b2">t</font> being   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="memstr_0.html#V4" title="MEMSTR_0:attr.4">-started</a>  <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a><br/>  for <font color="Olive" title="b3">Q</font> being   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  st  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Maroon" title="c4">s</font> holds <br/><font color="Olive" title="b1">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b2">t</font>,<font color="Olive" title="b3">Q</font> ) holds <br/><font color="Olive" title="b1">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">I</font> be   <a href="compos_1.html#NM5" title="COMPOS_1:NM.5">Program</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</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">t</font> being   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="memstr_0.html#V4" title="MEMSTR_0:attr.4">-started</a>  <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a><br/>  for <font color="Olive" title="b2">Q</font> being   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  st  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Maroon" title="c4">s</font> holds <br/><font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b1">t</font>,<font color="Olive" title="b2">Q</font> ) implies <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:6"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
 for <font color="Olive" title="b1">t</font> being   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="memstr_0.html#V4" title="MEMSTR_0:attr.4">-started</a>  <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a><br/>  for <font color="Olive" title="b2">Q</font> being   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  st  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Maroon" title="c4">s</font> holds <br/><font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b1">t</font>,<font color="Olive" title="b2">Q</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" title="c5">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">pI</font> =  <a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">sI</font> =  <a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c4">s</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">PI</font> = <font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</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> not  <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> <span class="p3">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span></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="c4">s</font></span>)</span>,$1)</span>)</span> <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="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span>;<br/>
<a NAME="E2:6"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>  holds  <font color="Maroon" title="c4">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">a</font> be    <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</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" title="c4">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">a</font></span><br/>

<a NAME="E1:6_1"/>
 not <font color="Maroon" title="c9">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="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_4.html#T18" title="SCMPDS_4:th.18">SCMPDS_4:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:6_1"/>
<font color="Maroon" title="c4">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T11" title="FUNCT_4:th.11">FUNCT_4:11</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><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E3:6"/>
 not <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c4">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"> contradiction</span><br/>

<a NAME="E4:6"/><span class="kw">then </span>
 ex <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>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_6.html#D2" title="SCMPDS_6:def.2">SCMPDS_6:def 2</a></span>;<br/>
<a NAME="E5:6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A4</font></span>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:6"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">n</font>]
 <span class="kw">and </span><br/><a NAME="E8:6"/><span class="lab"><font color="Green" title="E10">A6</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E5:6"><span class="lab"><font color="Green" title="E8">A4</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c10">n</font> = <font color="Maroon" title="c9">n</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">s2</font> =  <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</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> <span class="p2">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c4">s</font></span>)</span>,<font color="Maroon" title="c10">n</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c12">P2</font> = <font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">Ig</font> = (<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> <span class="p4">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span></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="c4">s</font></span>)</span>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p2">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c4">s</font></span>)</span>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span></span>)</span>) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (<span class="p1">(<span class="default"><a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">P0</font> = <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> <span class="p2">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c4">s</font></span>)</span>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</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> <span class="p0">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span></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="c4">s</font></span>)</span>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span></span>)</span>) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (<span class="p2">(<span class="default"><a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span>)</span>)</span> as   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">P1</font> = <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> <span class="p2">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c4">s</font></span>)</span>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</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> <span class="p0">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span></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="c4">s</font></span>)</span>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span></span>)</span>) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (<span class="p2">(<span class="default"><a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span>)</span>)</span> as   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">P3</font> = <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> <span class="p2">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</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> <span class="p4">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span></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="c4">s</font></span>)</span>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> 1</span>)</span>) as   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">P4</font> = <font color="Maroon" title="c16">P3</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p2">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c4">s</font></span>)</span>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span>) as   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ;<br/>
<a NAME="E14:6"/><span class="lab"><font color="Green" title="E11">U</font></span>: 
<font color="Maroon" title="c14">P0</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c17">P4</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T139" title="FUNCT_7:th.139">FUNCT_7:139</a></span>;<br/>
<a NAME="E15:6"/><span class="lab"><font color="Green" title="E12">A13</font></span>: 
 for <font color="Olive" title="b1">m</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>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">n</font> holds <br/> <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> <span class="p3">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span></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="c4">s</font></span>)</span>,<font color="Olive" title="b1">m</font>)</span>)</span> <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="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:6"><span class="lab"><font color="Green" title="E10">A6</font></span></a></span>;<br/>
<a NAME="E16:6"/><span class="lab"><font color="Green" title="E13">A17</font></span>: 
 <a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span>
 
<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/>
<a NAME="E17:6"/>
 <a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">P3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:6"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="funct_4.html#T25" title="FUNCT_4:th.25">FUNCT_4:25</a>, <a class="ref" href="funct_7.html#T89" title="FUNCT_7:th.89">FUNCT_7:89</a></span>;<br/>
<a NAME="E18:6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A18</font></span>: 
 <a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">P0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:6"><span class="lab"><font color="Green" title="E11">U</font></span></a>, <a class="txt" href="#E7:6"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="afinsq_1.html#T73" title="AFINSQ_1:th.73">AFINSQ_1:73</a>, <a class="ref" href="funct_7.html#T89" title="FUNCT_7:th.89">FUNCT_7:89</a></span>;<br/>
<a NAME="E19:6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">XX</font></span>: 
 <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c14">P0</font>,<span class="p1">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c4">s</font></span>)</span>,<font color="Maroon" title="c10">n</font>) <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</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> <span class="p2">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c4">s</font></span>)</span>,<font color="Maroon" title="c10">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:6"><span class="lab"><font color="Green" title="E13">A17</font></span></a>, <a class="txt" href="#E15:6"><span class="lab"><font color="Green" title="E12">A13</font></span></a>, <a class="ref" href="scmpds_4.html#T21" title="SCMPDS_4:th.21">SCMPDS_4:21</a></span>;<br/>
 <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c4">s</font></span>)</span> = 
 <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c4">s</font></span>)</span>

<br/>.= 
 <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Maroon" title="c4">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="scmpds_4.html#T8" title="SCMPDS_4:th.8">SCMPDS_4:8</a></span>
;<br/>
<a NAME="E21:6"/><span class="kw">then </span>
<font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a>  <a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c14">P0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>;<br/>
<a NAME="E22:6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A20</font></span>: 
<font color="Maroon" title="c14">P0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="extpro_1.html#R1" title="EXTPRO_1:pred.1">halts_on</a>  <a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c4">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_6.html#D3" title="SCMPDS_6:def.3">SCMPDS_6:def 3</a></span>;<br/>
<a NAME="E23:6"/><span class="lab"><font color="Green" title="E17">A21</font></span>: 
 not <font color="Maroon" title="c14">P0</font> <a href="extpro_1.html#R1" title="EXTPRO_1:pred.1">halts_on</a>  <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</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> <span class="p2">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c4">s</font></span>)</span>,<font color="Maroon" title="c10">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_4.html#T20" title="SCMPDS_4:th.20">SCMPDS_4:20</a></span>;<br/>
<a NAME="E24:6"/>
<font color="Maroon" title="c14">P0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="compos_1.html#K15" title="COMPOS_1:func.15">stop</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c14">P0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:6"><span class="lab"><font color="Green" title="E14">A18</font></span></a>, <a class="ref" href="funct_4.html#T98" title="FUNCT_4:th.98">FUNCT_4:98</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:6"><span class="lab"><font color="Green" title="E17">A21</font></span></a>, <a class="txt" href="#E19:6"><span class="lab"><font color="Green" title="E15">XX</font></span></a>, <a class="txt" href="#E22:6"><span class="lab"><font color="Green" title="E16">A20</font></span></a>, <a class="ref" href="extpro_1.html#T22" title="EXTPRO_1:th.22">EXTPRO_1:22</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/>


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