<?xml version="1.0"?>
<div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>let </b><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> ;<br/><b>let </b><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> ;<br/><b>let </b><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> ;<br/><b>let </b><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> ;<br/><b>assume </b><a NAME="E1:53_1"/><i><font color="Green" title="E39">A1</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="comput_1.html#K16" title="COMPUT_1:func.16">[+]</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> )
 ;<br/><b>then consider </b><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> <b> such that </b><br/><a NAME="E3:53_1"/><i><font color="Green" title="E40">A2</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="turing_1.html#T32" target="_self" title="TURING_1:th.32">Th32</a>, <a class="ref" href="finseq_2.html#T120" title="FINSEQ_2:th.120">FINSEQ_2:120</a></i>;<br/><a NAME="E4:53_1"/><i><font color="Green" title="E41">A3</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="turing_1.html#E1:53_1"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="turing_1.html#D15" target="_self" title="TURING_1:def.15">Def15</a></i>;<br/><a NAME="E5:53_1"/><i><font color="Green" title="E42">A4</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="turing_1.html#E3:53_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="finseq_1.html#T60" title="FINSEQ_1:th.60">FINSEQ_1:60</a></i>;<br/><a NAME="E6:53_1"/><i><font color="Green" title="E43">A5</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="turing_1.html#E1:53_1"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="turing_1.html#E3:53_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="finseq_1.html#T60" title="FINSEQ_1:th.60">FINSEQ_1:60</a></i>;<br/><a NAME="E7:53_1"/><i><font color="Green" title="E44">A6</font></i>: 
( <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> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">h1</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_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> )
 <b>by </b><i><a class="txt" href="turing_1.html#E1:53_1"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="turing_1.html#E4:53_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="turing_1.html#E5:53_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="turing_1.html#T31" target="_self" title="TURING_1:th.31">Th31</a></i>;<br/><b>thus </b><a NAME="E8:53_1"/>
<font color="Maroon" title="c1">s</font> is <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>
 <b>by </b><i><a class="txt" href="turing_1.html#E1:53_1"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="turing_1.html#E4:53_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="turing_1.html#E5:53_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="turing_1.html#T31" target="_self" title="TURING_1:th.31">Th31</a></i>;<br/><b>take </b><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>;<br/><b>take </b><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>;<br/><b>thus </b><a NAME="E9:53_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>
 <b>by </b><i><a class="txt" href="turing_1.html#E4:53_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="turing_1.html#E6:53_1"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="turing_1.html#T31" target="_self" title="TURING_1:th.31">Th31</a></i>;<br/><b>thus </b><a NAME="E10:53_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="comput_1.html#K16" title="COMPUT_1:func.16">[+]</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="turing_1.html#E3:53_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="comput_1.html#T89" title="COMPUT_1:th.89">COMPUT_1:89</a></i>;<br/><b>thus </b><a NAME="E11:53_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>
 <b>by </b><i><a class="txt" href="turing_1.html#E7:53_1"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="finseq_1.html#D9" title="FINSEQ_1:def.9">FINSEQ_1:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:53"/>
 <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#K16" title="COMPUT_1:func.16">[+]</a> 
 <b>by </b><i><a class="ref" href="turing_1.html#D16" target="_self" title="TURING_1:def.16">Def16</a></i>;<br/>


</div>
