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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c1">s</font> be   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">t</font> being   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a><br/>  for <font color="Olive" title="b2">h1</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> <br/>  for <font color="Olive" title="b3">x</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="comput_1.html#K15" title="COMPUT_1:func.15">[+]</a> &amp; <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>,<font color="Olive" title="b2">h1</font>,<font color="Olive" title="b1">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Olive" title="b1">t</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b2">h1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b3">x</font> holds <br/>( <font color="Maroon" title="c1">s</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp;  ex <font color="Olive" title="b4">h2</font>, <font color="Olive" title="b5">y</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="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">h2</font> &amp; <font color="Olive" title="b5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="comput_1.html#K15" title="COMPUT_1:func.15">[+]</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b4">h2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b5">y</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">t</font> be   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</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">h1</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> <br/>  for <font color="Olive" title="b2">x</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="comput_1.html#K15" title="COMPUT_1:func.15">[+]</a> &amp; <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>,<font color="Olive" title="b1">h1</font>,<font color="Maroon" title="c2">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Maroon" title="c2">t</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">h1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">x</font> holds <br/>( <font color="Maroon" title="c1">s</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp;  ex <font color="Olive" title="b3">h2</font>, <font color="Olive" title="b4">y</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="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">h2</font> &amp; <font color="Olive" title="b4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="comput_1.html#K15" title="COMPUT_1:func.15">[+]</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b3">h2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b4">y</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">h1</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">  for <font color="Olive" title="b1">x</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="comput_1.html#K15" title="COMPUT_1:func.15">[+]</a> &amp; <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>,<font color="Maroon" title="c3">h1</font>,<font color="Maroon" title="c2">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Maroon" title="c2">t</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">h1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">x</font> holds <br/>( <font color="Maroon" title="c1">s</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp;  ex <font color="Olive" title="b2">h2</font>, <font color="Olive" title="b3">y</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="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h2</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="comput_1.html#K15" title="COMPUT_1:func.15">[+]</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b2">h2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b3">y</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="comput_1.html#K15" title="COMPUT_1:func.15">[+]</a> &amp; <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>,<font color="Maroon" title="c3">h1</font>,<font color="Maroon" title="c2">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Maroon" title="c2">t</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">h1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">x</font> implies ( <font color="Maroon" title="c1">s</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp;  ex <font color="Olive" title="b1">h2</font>, <font color="Olive" title="b2">y</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="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h2</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="comput_1.html#K15" title="COMPUT_1:func.15">[+]</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">h2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:54_1"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="comput_1.html#K15" title="COMPUT_1:func.15">[+]</a>
 <span class="kw">and </span><br/><a NAME="E2:54_1"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
<font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"> the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>,<font color="Maroon" title="c3">h1</font>,<font color="Maroon" title="c2">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:54_1"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
<font color="Maroon" title="c2">t</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">h1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">s</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>  &amp;  ex <font color="Olive" title="b1">h2</font>, <font color="Olive" title="b2">y</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="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h2</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="comput_1.html#K15" title="COMPUT_1:func.15">[+]</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">h2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )</span><br/><a NAME="E4:54_1"/>
<font color="Maroon" title="c4">x</font> is   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of 2, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="turing_1.html#T28" target="_self" title="TURING_1:th.28">Th32</a>, <a class="ref" href="finseq_2.html#T131" title="FINSEQ_2:th.131">FINSEQ_2:131</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c6">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="E6:54_1"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T100" title="FINSEQ_2:th.100">FINSEQ_2:100</a></span>;<br/><a NAME="E7:54_1"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
<font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c3">h1</font>,<font color="Maroon" title="c2">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="turing_1.html#D14" target="_self" title="TURING_1:def.14">Def15</a></span>;<br/><a NAME="E8:54_1"/><span class="lab"><font color="Green" title="E43">A6</font></span>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">h1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">h1</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="finseq_1.html#T43" title="FINSEQ_1:th.43">FINSEQ_1:43</a></span>;<br/><span class="kw">hence </span><a NAME="E9:54_1"/>
<font color="Maroon" title="c1">s</font> is  <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E7:54_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="turing_1.html#T27" target="_self" title="TURING_1:th.27">Th31</a></span>; <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">h2</font>, <font color="Olive" title="b2">y</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="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h2</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="comput_1.html#K15" title="COMPUT_1:func.15">[+]</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">h2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">take </span><font color="Maroon" title="c7">h2</font> = 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">h1</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">y</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="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h2</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="comput_1.html#K15" title="COMPUT_1:func.15">[+]</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">h2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">y</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">take </span><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">j</font>; <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="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h2</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="comput_1.html#K15" title="COMPUT_1:func.15">[+]</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">h2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><a NAME="E10:54_1"/>
<font color="Maroon" title="c2">t</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">h1</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E6:54_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="finseq_1.html#T43" title="FINSEQ_1:th.43">FINSEQ_1:43</a></span>;<br/><span class="kw">hence </span><a NAME="E11:54_1"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:54_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="turing_1.html#T27" target="_self" title="TURING_1:th.27">Th31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="comput_1.html#K15" title="COMPUT_1:func.15">[+]</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">h2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">thus </span><a NAME="E12:54_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="comput_1.html#K15" title="COMPUT_1:func.15">[+]</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="comput_1.html#T85" title="COMPUT_1:th.85">COMPUT_1:85</a></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="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">h2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><a NAME="E13:54_1"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">h1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">j</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E7:54_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E8:54_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="ref" href="turing_1.html#T27" target="_self" title="TURING_1:th.27">Th31</a></span>;<br/><span class="kw">hence </span><a NAME="E14:54_1"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">h2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</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="E2:54"/>
 <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>  <a href="turing_1.html#R3" title="TURING_1:pred.3">computes</a>  <a href="comput_1.html#K15" title="COMPUT_1:func.15">[+]</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D15" target="_self" title="TURING_1:def.15">Def16</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>
