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<span class="kw">let </span><font color="Maroon" title="c2">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 <br/>  for <font color="Olive" title="b2">a</font> being   <a href="sf_mastr.html#NV2" title="SF_MASTR:NV.2">read-write</a>   <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> <br/>  for <font color="Olive" title="b3">s</font> being   <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>  st (  for <font color="Olive" title="b4">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Olive" title="b1">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b3">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">k</font>,<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">I</font>)</span>)</span> &amp; <font color="Olive" title="b1">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b3">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">k</font>,<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">I</font>)</span>)</span> ) ) &amp;  ex <font color="Olive" title="b4">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a>  the <a href="memstr_0.html#U1" title="MEMSTR_0:sel.1">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> st <br/> for <font color="Olive" title="b5">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( ( <font color="Olive" title="b4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b3">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b3">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b5">k</font></span>)</span> or <font color="Olive" title="b4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b3">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b5">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b3">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b5">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b3">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b5">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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b3">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b5">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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b3">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b5">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) holds <br/>(  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">I</font>) <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Olive" title="b3">s</font>,<font color="Maroon" title="c2">P</font> &amp;  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">I</font>) <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Olive" title="b3">s</font>,<font color="Maroon" title="c2">P</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">I</font> be   <a href="compos_1.html#NM5" title="COMPOS_1:NM.5">Program</a> of ; <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="sf_mastr.html#NV2" title="SF_MASTR:NV.2">read-write</a>   <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> <br/>  for <font color="Olive" title="b2">s</font> being   <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>  st (  for <font color="Olive" title="b3">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b2">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">k</font>,<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> &amp; <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b2">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">k</font>,<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> ) ) &amp;  ex <font color="Olive" title="b3">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a>  the <a href="memstr_0.html#U1" title="MEMSTR_0:sel.1">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> st <br/> for <font color="Olive" title="b4">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( ( <font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b2">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b2">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">k</font></span>)</span> or <font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b2">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b2">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b2">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <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="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b2">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <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="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b2">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) holds <br/>(  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Olive" title="b2">s</font>,<font color="Maroon" title="c2">P</font> &amp;  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Olive" title="b2">s</font>,<font color="Maroon" title="c2">P</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">a</font> be   <a href="sf_mastr.html#NV2" title="SF_MASTR:NV.2">read-write</a>   <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">  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="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  st (  for <font color="Olive" title="b2">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b1">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font>,<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> &amp; <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b1">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font>,<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> ) ) &amp;  ex <font color="Olive" title="b2">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a>  the <a href="memstr_0.html#U1" title="MEMSTR_0:sel.1">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> st <br/> for <font color="Olive" title="b3">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( ( <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b1">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b1">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">k</font></span>)</span> or <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b1">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b1">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b1">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b1">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b1">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) holds <br/>(  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c2">P</font> &amp;  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c2">P</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">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">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> &amp; <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> ) ) &amp;  ex <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a>  the <a href="memstr_0.html#U1" title="MEMSTR_0:sel.1">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> st <br/> for <font color="Olive" title="b2">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( ( <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> or <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) implies (  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">P</font> &amp;  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">P</font> ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:30"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> &amp; <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> )
 ; <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">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a>  the <a href="memstr_0.html#U1" title="MEMSTR_0:sel.1">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> not  for <font color="Olive" title="b2">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( ( <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> or <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) or (  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">P</font> &amp;  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">P</font> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">s1</font> =  <a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c5">s</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">P1</font> = <font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span>;<br/>
<a NAME="E2:30"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T93" title="FUNCT_4:th.93">FUNCT_4:93</a></span>;<br/>
<span class="kw">given </span><font color="Maroon" title="c8">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a>  the <a href="memstr_0.html#U1" title="MEMSTR_0:sel.1">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a><span class="kw"> such that </span><a NAME="E4:30"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( ( <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span> or <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <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="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <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="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">P</font> &amp;  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">P</font> )</span><br/>

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span>;<br/>
<a NAME="E5:30"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) or <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E28">A5</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:30"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E8:30"/><span class="lab"><font color="Green" title="E31">A8</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S17" title="NAT_1:sch.17">NAT_1:sch 17</a>(<a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E29">A6</font></span></a>);<br/></span>
<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>( $1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> implies  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 <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Olive" title="b1">k</font>) );<br/>
<div><span class="lab"><font color="Green" title="E32">A9</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">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="c10">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:30_1"/><span class="lab"><font color="Green" title="E33">A10</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">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="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c11">sk1</font> = <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c12">sk</font> = <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font>;<br/><span class="kw">assume </span><a NAME="E1:30_1_1"/><span class="lab"><font color="Green" title="E34">A11</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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 <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Olive" title="b1">m</font>)</span><br/><a NAME="E2:30_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E3:30_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:30_1_1"/><span class="kw">then </span>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">k</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E31">A8</font></span></a></span>;<br/><a NAME="E5:30_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A12</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E28">A5</font></span></a></span>;<br/><a NAME="E6:30_1_1"/><span class="lab"><font color="Green" title="E36">A13</font></span>: 
<font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E26">A1</font></span></a></span>;<br/><a NAME="E7:30_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">n</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> <span class="kw"> such that </span><br/><a NAME="E9:30_1_1"/><span class="lab"><font color="Green" title="E37">A14</font></span>: 
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c13">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c14">m</font> = <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="extpro_1.html#K8" title="EXTPRO_1:func.8">LifeSpan</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c3">I</font></span>)</span>,<span class="p3">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c14">m</font>)</span><br/><a NAME="E10:30_1_1"/><span class="lab"><font color="Green" title="E38">A15</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T93" title="FUNCT_4:th.93">FUNCT_4:93</a></span>;<br/><a NAME="E11:30_1_1"/>
( <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="extpro_1.html#K8" title="EXTPRO_1:func.8">LifeSpan</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c3">I</font></span>)</span>,<span class="p3">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span>) &amp; <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="scmfsa_9.html#D4" target="_self" title="SCMFSA_9:def.4">Def4</a></span>;<br/><a NAME="E12:30_1_1"/><span class="kw">then </span>
 <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1_1"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E5:30_1_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="scmfsa_9.html#T22" target="_self" title="SCMFSA_9:th.22">Th22</a>, <a class="txt" href="#E10:30_1_1"><span class="lab"><font color="Green" title="E38">A15</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E13:30_1_1"/>
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c14">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_1"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="scmfsa_9.html#T26" target="_self" title="SCMFSA_9:th.26">Th31</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">hence </span><a NAME="E3:30_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">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="E10:30"/><span class="lab"><font color="Green" title="E33">A16</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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 <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Olive" title="b1">k</font>)</span><br/>

<span class="kw">take </span><font color="Maroon" title="c10">n</font> = <span class="p1">(<span class="default"><a href="extpro_1.html#K8" title="EXTPRO_1:func.8">LifeSpan</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c3">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="c5">s</font></span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c10">n</font>)</span><br/>

<span class="kw">thus </span><a NAME="E2:30_2"/>
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c10">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa_9.html#T25" target="_self" title="SCMFSA_9:th.25">Th30</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>
<a NAME="E11:30"/><span class="lab"><font color="Green" title="E34">A17</font></span>: 
 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="#E10:30"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E9:30"><span class="lab"><font color="Green" title="E32">A9</font></span></a>);<br/></span>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_3_1"/>
( <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c9">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_3_1_1"/>
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">P</font> &amp;  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">P</font> )</span><br/><div class="add"><a NAME="E2:30_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <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="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E7:30"><span class="lab"><font color="Green" title="E30">A7</font></span></a></span>;<br/><a NAME="E3:30_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa_9.html#D4" target="_self" title="SCMFSA_9:def.4">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E4:30_3_1_1"/>
(  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">P</font> &amp;  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa_9.html#T18" target="_self" title="SCMFSA_9:th.18">Th18</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_3_1_2"/><span class="lab"><font color="Green" title="E35">A18</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">P</font> &amp;  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">P</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:30_3_1_2"/><span class="lab"><font color="Green" title="E36">A19</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">m</font> = <font color="Maroon" title="c9">m</font>, <font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c10">i</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="c13">sm</font> = <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font>;<br/><span class="kw">set </span><font color="Maroon" title="c14">si</font> = <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font>;<br/><a NAME="E5:30_3_1_2"/>
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_3_1_2"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:30_3_1_2"/><span class="kw">then </span>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">i</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E31">A8</font></span></a></span>;<br/><a NAME="E7:30_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A20</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E28">A5</font></span></a></span>;<br/><a NAME="E8:30_3_1_2"/><span class="lab"><font color="Green" title="E38">A21</font></span>: 
( <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> &amp; <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E26">A1</font></span></a></span>;<br/><a NAME="E9:30_3_1_2"/><span class="lab"><font color="Green" title="E39">XX</font></span>: 
 <a href="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</a>  <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"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="memstr_0.html#T2" title="MEMSTR_0:th.2">MEMSTR_0:2</a></span>;<br/><a NAME="E10:30_3_1_2"/>
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="extpro_1.html#K8" title="EXTPRO_1:func.8">LifeSpan</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c3">I</font></span>)</span>,<span class="p3">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_3_1_2"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="ref" href="scmfsa_9.html#D4" target="_self" title="SCMFSA_9:def.4">Def4</a></span>;<br/><a NAME="E11:30_3_1_2"/><span class="kw">then </span>
 <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_3_1_2"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="txt" href="#E7:30_3_1_2"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="ref" href="scmfsa_9.html#T22" target="_self" title="SCMFSA_9:th.22">Th22</a>, <a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E27">A3</font></span></a></span>;<br/><a NAME="E12:30_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font> is  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="memstr_0.html#V4" title="MEMSTR_0:attr.4">-started</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_3_1_2"><span class="lab"><font color="Green" title="E39">XX</font></span></a>, <a class="ref" href="memstr_0.html#D8" title="MEMSTR_0:def.8">MEMSTR_0:def 8</a></span>;<br/><a NAME="E13:30_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">B22</font></span>: 
 <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>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="memstr_0.html#T29" title="MEMSTR_0:th.29">MEMSTR_0:29</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c15">p</font> = <span class="p1">(<span class="default"><a href="extpro_1.html#K8" title="EXTPRO_1:func.8">LifeSpan</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c3">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="c5">s</font></span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3;<br/><span class="kw">set </span><font color="Maroon" title="c16">sm1</font> =  <a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font></span>)</span>;<br/><a NAME="E14:30_3_1_2"/>
<font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_3_1_2"><span class="lab"><font color="Green" title="E36">A19</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">n</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> <span class="kw"> such that </span><br/><a NAME="E16:30_3_1_2"/><span class="lab"><font color="Green" title="E41">A23</font></span>: 
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c17">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30"><span class="lab"><font color="Green" title="E34">A17</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">n</font> = <font color="Maroon" title="c17">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>  ;<br/><a NAME="E18:30_3_1_2"/><span class="lab"><font color="Green" title="E42">A27</font></span>: 
 <a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:30_3_1_2"><span class="lab"><font color="Green" title="E40">B22</font></span></a>, <a class="ref" href="funct_4.html#T98" title="FUNCT_4:th.98">FUNCT_4:98</a></span>;<br/><a NAME="E19:30_3_1_2"/><span class="lab"><font color="Green" title="E43">A28</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E7:30"><span class="lab"><font color="Green" title="E30">A7</font></span></a></span>;<br/><a NAME="E20:30_3_1_2"/><span class="kw">then </span>
 <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa_9.html#T18" target="_self" title="SCMFSA_9:th.18">Th18</a></span>;<br/><a NAME="E21:30_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa7b.html#D7" title="SCMFSA7B:def.7">SCMFSA7B:def 7</a></span>;<br/><a NAME="E22:30_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font></span>)</span>
 ;<br/><a NAME="E23:30_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font></span>)</span>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">j</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> <span class="kw"> such that </span><br/><a NAME="E25:30_3_1_2"/><span class="lab"><font color="Green" title="E44">A29</font></span>: 
 <a href="extpro_1.html#K3" title="EXTPRO_1:func.3">CurInstr</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:30_3_1_2"><span class="lab"><font color="Green" title="E42">A27</font></span></a>, <a class="ref" href="extpro_1.html#T29" title="EXTPRO_1:th.29">EXTPRO_1:29</a></span>;<br/><a NAME="E26:30_3_1_2"/><span class="lab"><font color="Green" title="E45">A30</font></span>: 
 <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">j</font></span>)</span>) <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c18">n</font>)</span>)</span>,<font color="Maroon" title="c19">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="extpro_1.html#T4" title="EXTPRO_1:th.4">EXTPRO_1:4</a></span>;<br/><a NAME="E27:30_3_1_2"/>
 <a href="extpro_1.html#K3" title="EXTPRO_1:func.3">CurInstr</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">j</font></span>)</span>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:30_3_1_2"><span class="lab"><font color="Green" title="E41">A23</font></span></a>, <a class="txt" href="#E25:30_3_1_2"><span class="lab"><font color="Green" title="E44">A29</font></span></a>, <a class="txt" href="#E26:30_3_1_2"><span class="lab"><font color="Green" title="E45">A30</font></span></a></span>;<br/><a NAME="E28:30_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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> <font color="Maroon" title="c5">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="extpro_1.html#T29" title="EXTPRO_1:th.29">EXTPRO_1:29</a></span>;<br/><span class="kw">hence </span><a NAME="E29:30_3_1_2"/>
 <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa7b.html#D7" title="SCMFSA7B:def.7">SCMFSA7B:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">P</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c20">q</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">  <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Olive">b<sub>1</sub></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> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span><br/><a NAME="E1:30_3_1_2_1"/><span class="lab"><font color="Green" title="E46">A31</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_3_1_2"><span class="lab"><font color="Green" title="E35">A18</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_3_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_3_1_2_1_1"/>
( <font color="Maroon" title="c20">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K8" title="EXTPRO_1:func.8">LifeSpan</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c3">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="c5">s</font></span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 or <font color="Maroon" title="c20">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K8" title="EXTPRO_1:func.8">LifeSpan</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c3">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="c5">s</font></span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_3_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E47">A32</font></span>: 
<font color="Maroon" title="c20">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K8" title="EXTPRO_1:func.8">LifeSpan</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c3">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="c5">s</font></span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Olive">b<sub>1</sub></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> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:30_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E48">A33</font></span>: 
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa_9.html#D4" target="_self" title="SCMFSA_9:def.4">Def4</a></span>;<br/><a NAME="E3:30_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A34</font></span>: 
( <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> &amp; <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E26">A1</font></span></a></span>;<br/><a NAME="E4:30_3_1_2_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="txt" href="#E1:30_3_1_2_1"><span class="lab"><font color="Green" title="E46">A31</font></span></a></span>;<br/><a NAME="E5:30_3_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E2:30_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E48">A33</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:30_3_1_2_1_1_1"/>
 <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c20">q</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> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E47">A32</font></span></a>, <a class="txt" href="#E3:30_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E49">A34</font></span></a>, <a class="ref" href="scmfsa_9.html#T22" target="_self" title="SCMFSA_9:th.22">Th22</a>, <a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E27">A3</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_3_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_3_1_2_1_1_2"/><span class="lab"><font color="Green" title="E47">A35</font></span>: 
<font color="Maroon" title="c20">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K8" title="EXTPRO_1:func.8">LifeSpan</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c3">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="c5">s</font></span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Olive">b<sub>1</sub></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> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span><br/><div class="add"><div><span class="lab"><font color="Green" title="E48">A36</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c21">k</font> = <span class="p1">(<span class="default"><a href="extpro_1.html#K8" title="EXTPRO_1:func.8">LifeSpan</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c3">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="c5">s</font></span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1 <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c21">k</font>) &amp; <font color="Maroon" title="c21">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">q</font> )</span><br/><span class="kw">thus </span><a NAME="E1:30_3_1_2_1_1_2_1"/>
( <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1 <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c21">k</font>) &amp; <font color="Maroon" title="c21">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E47">A35</font></span></a>, <a class="ref" href="scmfsa_9.html#T25" target="_self" title="SCMFSA_9:th.25">Th30</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">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font> &amp; $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  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 <br/>( <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Olive" title="b1">k</font>) &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">q</font> ) );<br/><a NAME="E3:30_3_1_2_1_1_2"/><span class="lab"><font color="Green" title="E49">A37</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font>
 ;<br/><a NAME="E4:30_3_1_2_1_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_3_1_2_1"><span class="lab"><font color="Green" title="E46">A31</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E5:30_3_1_2_1_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:30_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A38</font></span>: 
 ex <font color="Olive" title="b1">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">t</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E48">A36</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c21">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:30_3_1_2_1_1_2"/><span class="lab"><font color="Green" title="E51">A39</font></span>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">t</font>] &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">t</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="#E3:30_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E49">A37</font></span></a>, <a class="txt" href="#E6:30_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E50">A38</font></span></a>);<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c22">t</font> = <font color="Maroon" title="c21">t</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3_1_2_1_1_2_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_3_1_2_1_1_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_3_1_2_1_1_2_2_1"/>
( <font color="Maroon" title="c22">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">m</font> or <font color="Maroon" title="c22">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_3_1_2_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_3_1_2_1_1_2_2_1_1"/>
<font color="Maroon" title="c22">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c20">q</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> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">r</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> <span class="kw"> such that </span><br/><a NAME="E3:30_3_1_2_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E52">A40</font></span>: 
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c23">r</font>)
 <span class="kw">and </span><br/><a NAME="E4:30_3_1_2_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E53">A41</font></span>: 
<font color="Maroon" title="c23">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E51">A39</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c24">x</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:30_3_1_2_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E54">A42</font></span>: 
<font color="Maroon" title="c20">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">r</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_3_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E53">A41</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><a NAME="E7:30_3_1_2_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E55">A43</font></span>: 
 <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font>,<font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:30_3_1_2"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="ref" href="scmfsa_9.html#T18" target="_self" title="SCMFSA_9:th.18">Th18</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c25">x</font> = <font color="Maroon" title="c24">x</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/><a NAME="E9:30_3_1_2_1_1_2_2_1_1"/>
 <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c20">q</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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c25">x</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:30_3_1_2"><span class="lab"><font color="Green" title="E42">A27</font></span></a>, <a class="txt" href="#E3:30_3_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E52">A40</font></span></a>, <a class="txt" href="#E6:30_3_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E54">A42</font></span></a>, <a class="ref" href="extpro_1.html#T4" title="EXTPRO_1:th.4">EXTPRO_1:4</a></span>;<br/><span class="kw">hence </span><a NAME="E10:30_3_1_2_1_1_2_2_1_1"/>
 <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c20">q</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> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_3_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E55">A43</font></span></a>, <a class="ref" href="scmfsa7b.html#D6" title="SCMFSA7B:def.6">SCMFSA7B:def 6</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_3_1_2_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_3_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E52">A44</font></span>: 
<font color="Maroon" title="c22">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c20">q</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> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c23">Dt</font> = <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">t</font>;<br/><a NAME="E2:30_3_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E53">A45</font></span>: 
<font color="Maroon" title="c22">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E51">A39</font></span></a>, <a class="txt" href="#E1:30_3_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:30_3_1_2_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c22">t</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E31">A8</font></span></a></span>;<br/><a NAME="E4:30_3_1_2_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A46</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E28">A5</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c24">z</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> <span class="kw"> such that </span><br/><a NAME="E6:30_3_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E55">A47</font></span>: 
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">t</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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c24">z</font>)
 <span class="kw">and </span><br/><a NAME="E7:30_3_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E56">A48</font></span>: 
<font color="Maroon" title="c24">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E51">A39</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c25">z2</font> = <font color="Maroon" title="c24">z</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="extpro_1.html#K8" title="EXTPRO_1:func.8">LifeSpan</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c3">I</font></span>)</span>,<span class="p3">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">t</font></span>)</span></span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span>;<br/><span class="kw">consider </span><font color="Maroon" title="c26">w</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E9:30_3_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E57">A49</font></span>: 
<font color="Maroon" title="c20">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">z</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_3_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E56">A48</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><a NAME="E10:30_3_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E58">A50</font></span>: 
( <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">t</font>,<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> &amp; <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">t</font>,<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E26">A1</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c27">y</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E12:30_3_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E59">A51</font></span>: 
<font color="Maroon" title="c22">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">y</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E51">A39</font></span></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c28">y</font> = <font color="Maroon" title="c27">y</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="c29">Dy</font> = <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c28">y</font>;<br/><a NAME="E14:30_3_1_2_1_1_2_2_1_2"/>
<font color="Maroon" title="c28">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:30_3_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E59">A51</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E15:30_3_1_2_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c28">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E51">A39</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E16:30_3_1_2_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c28">y</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E31">A8</font></span></a></span>;<br/><a NAME="E17:30_3_1_2_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A52</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c28">y</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E28">A5</font></span></a></span>;<br/><a NAME="E18:30_3_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E61">A53</font></span>: 
( <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c28">y</font>,<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> &amp; <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c28">y</font>,<font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E26">A1</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c30">w</font> = <font color="Maroon" title="c26">w</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/><a NAME="E20:30_3_1_2_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">t</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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c28">y</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="extpro_1.html#K8" title="EXTPRO_1:func.8">LifeSpan</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c3">I</font></span>)</span>,<span class="p3">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c28">y</font></span>)</span></span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:30_3_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E59">A51</font></span></a>, <a class="ref" href="scmfsa_9.html#D4" target="_self" title="SCMFSA_9:def.4">Def4</a></span>;<br/><a NAME="E21:30_3_1_2_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A54</font></span>: 
 <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:30_3_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E61">A53</font></span></a>, <a class="txt" href="#E17:30_3_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E60">A52</font></span></a>, <a class="ref" href="scmfsa_9.html#T22" target="_self" title="SCMFSA_9:th.22">Th22</a>, <a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E27">A3</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3_1_2_1_1_2_2_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:30_3_1_2_1_1_2_2_1_2_1"/><span class="lab"><font color="Green" title="E63">A55</font></span>: 
<font color="Maroon" title="c24">z</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="extpro_1.html#K8" title="EXTPRO_1:func.8">LifeSpan</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c3">I</font></span>)</span>,<span class="p3">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">t</font></span>)</span></span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">q</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/><div><span class="lab"><font color="Green" title="E64">A56</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3_1_2_1_1_2_2_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c31">k</font> = <font color="Maroon" title="c24">z</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="extpro_1.html#K8" title="EXTPRO_1:func.8">LifeSpan</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c3">I</font></span>)</span>,<span class="p3">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">t</font></span>)</span></span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c31">k</font>) &amp; <font color="Maroon" title="c31">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">q</font> )</span><br/><span class="kw">thus </span><a NAME="E1:30_3_1_2_1_1_2_2_1_2_1_1"/>
( <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c31">k</font>) &amp; <font color="Maroon" title="c31">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_3_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E55">A47</font></span></a>, <a class="txt" href="#E21:30_3_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E62">A54</font></span></a>, <a class="txt" href="#E1:30_3_1_2_1_1_2_2_1_2_1"><span class="lab"><font color="Green" title="E63">A55</font></span></a>, <a class="ref" href="scmfsa_9.html#T26" target="_self" title="SCMFSA_9:th.26">Th31</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><a NAME="E3:30_3_1_2_1_1_2_2_1_2_1"/>
<font color="Maroon" title="c22">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_3_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:30_3_1_2_1_1_2_2_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E51">A39</font></span></a>, <a class="txt" href="#E2:30_3_1_2_1_1_2_2_1_2_1"><span class="lab"><font color="Green" title="E64">A56</font></span></a>, <a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</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><a NAME="E23:30_3_1_2_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A57</font></span>: 
<font color="Maroon" title="c30">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K8" title="EXTPRO_1:func.8">LifeSpan</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c3">I</font></span>)</span>,<span class="p2">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">t</font></span>)</span></span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_3_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E57">A49</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E24:30_3_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E64">A58</font></span>: 
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">t</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">t</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_3_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E55">A47</font></span></a>, <a class="txt" href="#E21:30_3_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E62">A54</font></span></a>, <a class="ref" href="scmfsa_9.html#T26" target="_self" title="SCMFSA_9:th.26">Th31</a></span>;<br/><a NAME="E25:30_3_1_2_1_1_2_2_1_2"/>
 <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c20">q</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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<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="c30">w</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:30_3_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E64">A58</font></span></a>, <a class="txt" href="#E6:30_3_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E55">A47</font></span></a>, <a class="txt" href="#E9:30_3_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E57">A49</font></span></a>, <a class="ref" href="extpro_1.html#T4" title="EXTPRO_1:th.4">EXTPRO_1:4</a></span>;<br/><span class="kw">hence </span><a NAME="E26:30_3_1_2_1_1_2_2_1_2"/>
 <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c20">q</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> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:30_3_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E63">A57</font></span></a>, <a class="txt" href="#E10:30_3_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E58">A50</font></span></a>, <a class="txt" href="#E4:30_3_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="ref" href="scmfsa_9.html#T22" target="_self" title="SCMFSA_9:th.22">Th22</a>, <a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E27">A3</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E11:30_3_1_2_1_1_2"/>
 <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="c2">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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="c5">s</font></span>)</span>,<font color="Maroon" title="c20">q</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> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>)</span>)</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E31:30_3_1_2"/>
 <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa7b.html#D6" title="SCMFSA7B:def.6">SCMFSA7B:def 6</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:30"/>
(  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">P</font> &amp;  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">P</font> )
 ; <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>
