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<b>let </b><font color="Maroon" title="c1">n</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>assume </b><a NAME="E1:78"/><i><font color="Green" title="E58">A1</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">s</font> be   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <a href="turing_1.html#K20" title="TURING_1:func.20">U3(n)Turing</a> ;<br/><b>let </b><font color="Maroon" title="c3">t</font> be   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <a href="turing_1.html#K20" title="TURING_1:func.20">U3(n)Turing</a> ;<br/><b>let </b><font color="Maroon" title="c4">h</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="c5">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>set </b><font color="Maroon" title="c6">pj</font> = <font color="Maroon" title="c1">n</font> <a href="comput_1.html#K7" title="COMPUT_1:func.7">proj</a> 3;<br/><b>assume </b><a NAME="E1:78_1"/><i><font color="Green" title="E59">A2</font></i>: 
( <font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="comput_1.html#K7" title="COMPUT_1:func.7">proj</a> 3</span>)</span> &amp; <font color="Maroon" title="c2">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#K20" title="TURING_1:func.20">U3(n)Turing</a> ,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Maroon" title="c3">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="c4">h</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="c5">x</font> )
 ;<br/><a NAME="E2:78_1"/>
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="comput_1.html#K7" title="COMPUT_1:func.7">proj</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T41" title="COMPUT_1:th.41">COMPUT_1:41</a></i>;<br/><a NAME="E3:78_1"/><b>then </b><i><font color="Green" title="E60">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="comput_1.html#K7" title="COMPUT_1:func.7">proj</a> 3</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#T24" title="COMPUT_1:th.24">COMPUT_1:24</a></i>;<br/><a NAME="E4:78_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="turing_1.html#E1:78_1"><i><font color="Green" title="E59">A2</font></i></a></i>;<br/><a NAME="E5:78_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="finseq_2.html#D4" title="FINSEQ_2:def.4">FINSEQ_2:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">f</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E7:78_1"/><i><font color="Green" title="E61">A4</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 ;<br/><a NAME="E8:78_1"/><i><font color="Green" title="E62">A5</font></i>: 
<font color="Maroon" title="c2">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">0,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">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#E1:78_1"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="turing_1.html#D22" target="_self" title="TURING_1:def.22">Def22</a></i>;<br/><b>hence </b><a NAME="E9:78_1"/>
<font color="Maroon" title="c2">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#E1:78"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="turing_1.html#E1:78_1"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="turing_1.html#E7:78_1"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="turing_1.html#T42" target="_self" title="TURING_1:th.42">Th42</a></i>;<br/><b>take </b><font color="Maroon" title="c8">h2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4;<br/><b>take </b><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c5">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 3;<br/><b>thus </b><a NAME="E10:78_1"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c2">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="c8">h2</font>
 <b>by </b><i><a class="txt" href="turing_1.html#E1:78"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="turing_1.html#E1:78_1"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="turing_1.html#E7:78_1"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="turing_1.html#E8:78_1"><i><font color="Green" title="E62">A5</font></i></a>, <a class="ref" href="turing_1.html#T42" target="_self" title="TURING_1:th.42">Th42</a></i>;<br/><b>thus </b><font color="Maroon" title="c9">y</font> = 
<font color="Maroon" title="c5">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3
<b>by </b><i><a class="txt" href="turing_1.html#E1:78"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="turing_1.html#E7:78_1"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="comput_1.html#K7" title="COMPUT_1:func.7">proj</a> 3</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="txt" href="turing_1.html#E1:78_1"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="turing_1.html#E3:78_1"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="comput_1.html#T42" title="COMPUT_1:th.42">COMPUT_1:42</a></i>
;<br/><a NAME="E12:78_1"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c2">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"><font color="Maroon" title="c8">h2</font>,<font color="Maroon" title="c9">y</font></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#E1:78"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="turing_1.html#E1:78_1"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="turing_1.html#E7:78_1"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="turing_1.html#E8:78_1"><i><font color="Green" title="E62">A5</font></i></a>, <a class="ref" href="turing_1.html#T42" target="_self" title="TURING_1:th.42">Th42</a></i>;<br/><b>hence </b><a NAME="E13:78_1"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c2">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="c8">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="c9">y</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><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="E3:78"/>
 <a href="turing_1.html#K20" title="TURING_1:func.20">U3(n)Turing</a>  <a href="turing_1.html#R3" title="TURING_1:pred.3">computes</a> <font color="Maroon" title="c1">n</font> <a href="comput_1.html#K7" title="COMPUT_1:func.7">proj</a> 3
 <b>by </b><i><a class="ref" href="turing_1.html#D16" target="_self" title="TURING_1:def.16">Def16</a></i>;<br/>


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