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

<b>let </b><font color="Maroon" title="c1">tm1</font>, <font color="Maroon" title="c2">tm2</font> be    <a href="turing_1.html#L1" title="TURING_1:struct.1">TuringStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">s1</font> be   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <font color="Maroon" title="c1">tm1</font>;<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">t</font> be   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <font color="Maroon" title="c1">tm1</font>;<br/><b>let </b><font color="Maroon" title="c6">s2</font> be   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <font color="Maroon" title="c2">tm2</font>;<br/><b>let </b><font color="Maroon" title="c7">s3</font> be   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K32" title="TURING_1:func.32">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>;<br/>











<b>set </b><font color="Maroon" title="c8">p0</font> = the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c1">tm1</font>;<br/>
<b>set </b><font color="Maroon" title="c9">q0</font> = the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font>;<br/>
<b>assume </b><a NAME="E1:95"/><i><font color="Green" title="E70">A1</font></i>: 
( <font color="Maroon" title="c3">s1</font> is <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a> &amp; <font color="Maroon" title="c3">s1</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 <font color="Maroon" title="c1">tm1</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Maroon" title="c6">s2</font> is <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a> &amp; <font color="Maroon" title="c6">s2</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 <font color="Maroon" title="c2">tm2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Maroon" title="c7">s3</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 <span class="p2">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K32" title="TURING_1:func.32">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> )
 ;<br/>

<b>set </b><font color="Maroon" title="c10">pF</font> = the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>;<br/>
<b>set </b><font color="Maroon" title="c11">qF</font> = the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c2">tm2</font>;<br/>
<b>consider </b><font color="Maroon" title="c12">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> <b> such that </b><br/><a NAME="E3:95"/><i><font color="Green" title="E71">A2</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font> &amp;  <a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font> ) )
 <b>by </b><i><a class="txt" href="turing_1.html#E1:95"><i><font color="Green" title="E70">A1</font></i></a>, <a class="ref" href="turing_1.html#T16" target="_self" title="TURING_1:th.16">Th16</a></i>;<br/>
<a NAME="E4:95"/><i><font color="Green" title="E72">A3</font></i>: 
<font color="Maroon" title="c7">s3</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"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c1">tm1</font>,the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c5">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#E1:95"><i><font color="Green" title="E70">A1</font></i></a>, <a class="ref" href="turing_1.html#D32" target="_self" title="TURING_1:def.32">Def32</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> implies ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>,the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  ) );<br/>
<a NAME="E5:95"/><i><font color="Green" title="E73">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:95_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font>
 ;<br/>

<i><font color="Green" title="E74">A5</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  = 
<font color="Maroon" title="c3">s1</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>
<br/>.= 
the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c1">tm1</font>
<b>by </b><i><a class="txt" href="turing_1.html#E1:95"><i><font color="Green" title="E70">A1</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  = 
<font color="Maroon" title="c7">s3</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c1">tm1</font>,the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E4:95"><i><font color="Green" title="E72">A3</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>
;<br/>
<b>hence </b><a NAME="E4:95_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>,the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
 <b>by </b><i><a class="txt" href="turing_1.html#E2:95_1"><i><font color="Green" title="E74">A5</font></i></a></i>;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  = 
<font color="Maroon" title="c3">s1</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>
<br/>.= 
<font color="Maroon" title="c4">h</font>
<b>by </b><i><a class="txt" href="turing_1.html#E1:95"><i><font color="Green" title="E70">A1</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>
<br/>.= 
<font color="Maroon" title="c7">s3</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
<b>by </b><i><a class="txt" href="turing_1.html#E1:95"><i><font color="Green" title="E70">A1</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  = 
<font color="Maroon" title="c3">s1</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> 
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>
<br/>.= 
<font color="Maroon" title="c5">t</font>
<b>by </b><i><a class="txt" href="turing_1.html#E1:95"><i><font color="Green" title="E70">A1</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>
<br/>.= 
<font color="Maroon" title="c7">s3</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> 
<b>by </b><i><a class="txt" href="turing_1.html#E1:95"><i><font color="Green" title="E70">A1</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> 
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E6:95"/><i><font color="Green" title="E74">A6</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">i</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="E1:95_2"/><i><font color="Green" title="E75">A7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">i</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_2_1"/><i><font color="Green" title="E76">A8</font></i>: 
<font color="Maroon" title="c13">i</font> <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="c12">k</font>
 ;<br/><b>set </b><font color="Maroon" title="c14">s1i1</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon" title="c15">s1i</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>;<br/><b>set </b><font color="Maroon" title="c16">s3i1</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon" title="c17">s3i</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>;<br/><a NAME="E2:95_2_1"/><i><font color="Green" title="E77">A9</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E3:95_2_1"/><b>then </b><i><font color="Green" title="E78">A10</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k</font>
 <b>by </b><i><a class="txt" href="turing_1.html#E1:95_2_1"><i><font color="Green" title="E76">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">ss1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  as   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <font color="Maroon" title="c1">tm1</font> ;<br/><b>reconsider </b><font color="Maroon" title="c19">ss3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  as   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K32" title="TURING_1:func.32">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c20">h</font> =  <a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c21">y</font> = <font color="Maroon" title="c18">ss1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">h</font> as   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c1">tm1</font> ;<br/><b>set </b><font color="Maroon" title="c22">p</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> ;<br/><b>set </b><font color="Maroon" title="c23">g</font> =  <a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span>;<br/><a NAME="E8:95_2_1"/><i><font color="Green" title="E79">A11</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>
 <b>by </b><i><a class="txt" href="turing_1.html#E3:95"><i><font color="Green" title="E71">A2</font></i></a>, <a class="txt" href="turing_1.html#E3:95_2_1"><i><font color="Green" title="E78">A10</font></i></a></i>;<br/><a NAME="E9:95_2_1"/><i><font color="Green" title="E80">A12</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K32" title="TURING_1:func.32">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:95_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K32" title="TURING_1:func.32">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>
 ;<br/>

<a NAME="E2:95_2_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>,the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="turing_1.html#E1:95_2"><i><font color="Green" title="E75">A7</font></i></a>, <a class="txt" href="turing_1.html#E1:95_2_1"><i><font color="Green" title="E76">A8</font></i></a>, <a class="txt" href="turing_1.html#E2:95_2_1"><i><font color="Green" title="E77">A9</font></i></a>, <a class="ref" href="turing_1.html#D32" target="_self" title="TURING_1:def.32">Def32</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E3:95_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="turing_1.html#E8:95_2_1"><i><font color="Green" title="E79">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon" title="c24">f</font> =  <a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span>;<br/><i><font color="Green" title="E81">A13</font></i>:  <a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> = 
the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K32" title="TURING_1:func.32">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>,the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c21">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:95_2"><i><font color="Green" title="E75">A7</font></i></a>, <a class="txt" href="turing_1.html#E1:95_2_1"><i><font color="Green" title="E76">A8</font></i></a>, <a class="txt" href="turing_1.html#E2:95_2_1"><i><font color="Green" title="E77">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>,the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span></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#E8:95_2_1"><i><font color="Green" title="E79">A11</font></i></a>, <a class="ref" href="turing_1.html#T48" target="_self" title="TURING_1:th.48">Th48</a></i>
;<br/><a NAME="E11:95_2_1"/><b>then </b><i><font color="Green" title="E82">A14</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></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> <span class="p1">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
 <b>by </b><i><a class="ref" href="mcart_1.html#D6" title="MCART_1:def.6">MCART_1:def 6</a></i>;<br/><i><font color="Green" title="E83">A15</font></i>: <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span>
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c18">ss1</font>,<font color="Maroon" title="c20">h</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E8:95_2_1"><i><font color="Green" title="E79">A11</font></i></a>, <a class="ref" href="turing_1.html#D7" target="_self" title="TURING_1:def.7">Def7</a></i>
;<br/><i><font color="Green" title="E84">A16</font></i>: <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span>
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c19">ss3</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E9:95_2_1"><i><font color="Green" title="E80">A12</font></i></a>, <a class="ref" href="turing_1.html#D7" target="_self" title="TURING_1:def.7">Def7</a></i>
;<br/><b>thus </b><span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>,the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> = 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>,the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E12:95_2_1"><i><font color="Green" title="E83">A15</font></i></a>, <a class="ref" href="mcart_1.html#D5" title="MCART_1:def.5">MCART_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
<b>by </b><i><a class="txt" href="turing_1.html#E10:95_2_1"><i><font color="Green" title="E81">A13</font></i></a>, <a class="ref" href="mcart_1.html#D5" title="MCART_1:def.5">MCART_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
<b>by </b><i><a class="txt" href="turing_1.html#E13:95_2_1"><i><font color="Green" title="E84">A16</font></i></a>, <a class="ref" href="mcart_1.html#D5" title="MCART_1:def.5">MCART_1:def 5</a></i>
;<br/><a NAME="E15:95_2_1"/>
 <a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="turing_1.html#E10:95_2_1"><i><font color="Green" title="E81">A13</font></i></a>, <a class="ref" href="mcart_1.html#D7" title="MCART_1:def.7">MCART_1:def 7</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  = 
<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:95_2"><i><font color="Green" title="E75">A7</font></i></a>, <a class="txt" href="turing_1.html#E1:95_2_1"><i><font color="Green" title="E76">A8</font></i></a>, <a class="txt" href="turing_1.html#E2:95_2_1"><i><font color="Green" title="E77">A9</font></i></a>, <a class="txt" href="turing_1.html#E12:95_2_1"><i><font color="Green" title="E83">A15</font></i></a>, <a class="ref" href="mcart_1.html#D6" title="MCART_1:def.6">MCART_1:def 6</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
<b>by </b><i><a class="txt" href="turing_1.html#E13:95_2_1"><i><font color="Green" title="E84">A16</font></i></a>, <a class="ref" href="mcart_1.html#D6" title="MCART_1:def.6">MCART_1:def 6</a></i>
;<br/><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  = 
<font color="Maroon" title="c19">ss3</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="turing_1.html#K2" title="TURING_1:func.2">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E12:95_2_1"><i><font color="Green" title="E83">A15</font></i></a>, <a class="ref" href="mcart_1.html#D7" title="MCART_1:def.7">MCART_1:def 7</a>, <a class="txt" href="turing_1.html#E1:95_2"><i><font color="Green" title="E75">A7</font></i></a>, <a class="txt" href="turing_1.html#E1:95_2_1"><i><font color="Green" title="E76">A8</font></i></a>, <a class="txt" href="turing_1.html#E2:95_2_1"><i><font color="Green" title="E77">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> 
<b>by </b><i><a class="txt" href="turing_1.html#E1:95_2"><i><font color="Green" title="E75">A7</font></i></a>, <a class="txt" href="turing_1.html#E1:95_2_1"><i><font color="Green" title="E76">A8</font></i></a>, <a class="txt" href="turing_1.html#E2:95_2_1"><i><font color="Green" title="E77">A9</font></i></a>, <a class="txt" href="turing_1.html#E11:95_2_1"><i><font color="Green" title="E82">A14</font></i></a>, <a class="txt" href="turing_1.html#E13:95_2_1"><i><font color="Green" title="E84">A16</font></i></a>, <a class="ref" href="mcart_1.html#D7" title="MCART_1:def.7">MCART_1:def 7</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:95_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c13">s1k</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font>;<br/>
<b>set </b><font color="Maroon" title="c14">s3k</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font>;<br/>
<a NAME="E7:95"/>
for <font color="Olive" title="b1">i</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">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="turing_1.html#E5:95"><i><font color="Green" title="E73">A4</font></i></a>, <a class="txt" href="turing_1.html#E6:95"><i><font color="Green" title="E74">A6</font></i></a>);<br/></i>
<a NAME="E8:95"/><b>then </b><i><font color="Green" title="E75">A17</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>,the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  )
 
;<br/>
<b>consider </b><font color="Maroon" title="c15">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> <b> such that </b><br/><a NAME="E10:95"/><i><font color="Green" title="E76">A18</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c2">tm2</font> &amp;  <a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">m</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">m</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c2">tm2</font> ) )
 <b>by </b><i><a class="txt" href="turing_1.html#E1:95"><i><font color="Green" title="E70">A1</font></i></a>, <a class="ref" href="turing_1.html#T16" target="_self" title="TURING_1:th.16">Th16</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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>means </b>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font> implies ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  ) );<br/>
<a NAME="E11:95"/><i><font color="Green" title="E77">A19</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:95_3"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font>
 ;<br/>

<b>thus </b><span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> = 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,the <a href="turing_1.html#U4" title="TURING_1:sel.4">InitS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:95"><i><font color="Green" title="E70">A1</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
<b>by </b><i><a class="txt" href="turing_1.html#E3:95"><i><font color="Green" title="E71">A2</font></i></a>, <a class="txt" href="turing_1.html#E8:95"><i><font color="Green" title="E75">A17</font></i></a>, <a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  = 
<font color="Maroon" title="c6">s2</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
<b>by </b><i><a class="txt" href="turing_1.html#E1:95"><i><font color="Green" title="E70">A1</font></i></a>, <a class="txt" href="turing_1.html#E3:95"><i><font color="Green" title="E71">A2</font></i></a>, <a class="txt" href="turing_1.html#E8:95"><i><font color="Green" title="E75">A17</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  = 
<font color="Maroon" title="c6">s2</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> 
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> 
<b>by </b><i><a class="txt" href="turing_1.html#E1:95"><i><font color="Green" title="E70">A1</font></i></a>, <a class="txt" href="turing_1.html#E3:95"><i><font color="Green" title="E71">A2</font></i></a>, <a class="txt" href="turing_1.html#E8:95"><i><font color="Green" title="E75">A17</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> 
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E12:95"/><i><font color="Green" title="E78">A20</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">i</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="E1:95_4"/><i><font color="Green" title="E79">A21</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">i</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_4_1"/><i><font color="Green" title="E80">A22</font></i>: 
<font color="Maroon" title="c16">i</font> <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="c15">m</font>
 ;<br/><b>set </b><font color="Maroon" title="c17">s2i1</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon" title="c18">s2i</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font>;<br/><b>set </b><font color="Maroon" title="c19">ski1</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon" title="c20">ski</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font>;<br/><a NAME="E2:95_4_1"/><i><font color="Green" title="E81">A23</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E3:95_4_1"/><b>then </b><i><font color="Green" title="E82">A24</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">m</font>
 <b>by </b><i><a class="txt" href="turing_1.html#E1:95_4_1"><i><font color="Green" title="E80">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">ss2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  as   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <font color="Maroon" title="c2">tm2</font> ;<br/><b>reconsider </b><font color="Maroon" title="c22">ssk</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  as   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K32" title="TURING_1:func.32">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c23">h</font> =  <a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c24">y</font> = <font color="Maroon" title="c21">ss2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">h</font> as   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c2">tm2</font> ;<br/><b>set </b><font color="Maroon" title="c25">q</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> ;<br/><b>set </b><font color="Maroon" title="c26">g</font> =  <a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span>;<br/><a NAME="E8:95_4_1"/><i><font color="Green" title="E83">A25</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c2">tm2</font>
 <b>by </b><i><a class="txt" href="turing_1.html#E10:95"><i><font color="Green" title="E76">A18</font></i></a>, <a class="txt" href="turing_1.html#E3:95_4_1"><i><font color="Green" title="E82">A24</font></i></a></i>;<br/><a NAME="E9:95_4_1"/><i><font color="Green" title="E84">A26</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K32" title="TURING_1:func.32">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_4_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:95_4_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K32" title="TURING_1:func.32">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>
 ;<br/>

<a NAME="E2:95_4_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c2">tm2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="turing_1.html#E1:95_4"><i><font color="Green" title="E79">A21</font></i></a>, <a class="txt" href="turing_1.html#E1:95_4_1"><i><font color="Green" title="E80">A22</font></i></a>, <a class="txt" href="turing_1.html#E2:95_4_1"><i><font color="Green" title="E81">A23</font></i></a>, <a class="ref" href="turing_1.html#D32" target="_self" title="TURING_1:def.32">Def32</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E3:95_4_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="turing_1.html#E8:95_4_1"><i><font color="Green" title="E83">A25</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon" title="c27">f</font> =  <a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span>;<br/><i><font color="Green" title="E85">A27</font></i>:  <a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span> = 
the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K32" title="TURING_1:func.32">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c24">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:95_4"><i><font color="Green" title="E79">A21</font></i></a>, <a class="txt" href="turing_1.html#E1:95_4_1"><i><font color="Green" title="E80">A22</font></i></a>, <a class="txt" href="turing_1.html#E2:95_4_1"><i><font color="Green" title="E81">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#T49" target="_self" title="TURING_1:th.49">Th49</a></i>
;<br/><a NAME="E11:95_4_1"/><b>then </b><i><font color="Green" title="E86">A28</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span></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> <span class="p1">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
 <b>by </b><i><a class="ref" href="mcart_1.html#D6" title="MCART_1:def.6">MCART_1:def 6</a></i>;<br/><i><font color="Green" title="E87">A29</font></i>: <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span>
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c21">ss2</font>,<font color="Maroon" title="c23">h</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E8:95_4_1"><i><font color="Green" title="E83">A25</font></i></a>, <a class="ref" href="turing_1.html#D7" target="_self" title="TURING_1:def.7">Def7</a></i>
;<br/><i><font color="Green" title="E88">A30</font></i>: <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span>
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c22">ssk</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E9:95_4_1"><i><font color="Green" title="E84">A26</font></i></a>, <a class="ref" href="turing_1.html#D7" target="_self" title="TURING_1:def.7">Def7</a></i>
;<br/><b>thus </b><span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> = 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E12:95_4_1"><i><font color="Green" title="E87">A29</font></i></a>, <a class="ref" href="mcart_1.html#D5" title="MCART_1:def.5">MCART_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
<b>by </b><i><a class="txt" href="turing_1.html#E10:95_4_1"><i><font color="Green" title="E85">A27</font></i></a>, <a class="ref" href="mcart_1.html#D5" title="MCART_1:def.5">MCART_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
<b>by </b><i><a class="txt" href="turing_1.html#E13:95_4_1"><i><font color="Green" title="E88">A30</font></i></a>, <a class="ref" href="mcart_1.html#D5" title="MCART_1:def.5">MCART_1:def 5</a></i>
;<br/><a NAME="E15:95_4_1"/>
 <a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="turing_1.html#E10:95_4_1"><i><font color="Green" title="E85">A27</font></i></a>, <a class="ref" href="mcart_1.html#D7" title="MCART_1:def.7">MCART_1:def 7</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  = 
<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="turing_1.html#K6" title="TURING_1:func.6">offset</a> <span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:95_4"><i><font color="Green" title="E79">A21</font></i></a>, <a class="txt" href="turing_1.html#E1:95_4_1"><i><font color="Green" title="E80">A22</font></i></a>, <a class="txt" href="turing_1.html#E2:95_4_1"><i><font color="Green" title="E81">A23</font></i></a>, <a class="txt" href="turing_1.html#E12:95_4_1"><i><font color="Green" title="E87">A29</font></i></a>, <a class="ref" href="mcart_1.html#D6" title="MCART_1:def.6">MCART_1:def 6</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
<b>by </b><i><a class="txt" href="turing_1.html#E13:95_4_1"><i><font color="Green" title="E88">A30</font></i></a>, <a class="ref" href="mcart_1.html#D6" title="MCART_1:def.6">MCART_1:def 6</a></i>
;<br/><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  = 
<font color="Maroon" title="c22">ssk</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">h</font> <a href="turing_1.html#K2" title="TURING_1:func.2">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E12:95_4_1"><i><font color="Green" title="E87">A29</font></i></a>, <a class="ref" href="mcart_1.html#D7" title="MCART_1:def.7">MCART_1:def 7</a>, <a class="txt" href="turing_1.html#E1:95_4"><i><font color="Green" title="E79">A21</font></i></a>, <a class="txt" href="turing_1.html#E1:95_4_1"><i><font color="Green" title="E80">A22</font></i></a>, <a class="txt" href="turing_1.html#E2:95_4_1"><i><font color="Green" title="E81">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> 
<b>by </b><i><a class="txt" href="turing_1.html#E1:95_4"><i><font color="Green" title="E79">A21</font></i></a>, <a class="txt" href="turing_1.html#E1:95_4_1"><i><font color="Green" title="E80">A22</font></i></a>, <a class="txt" href="turing_1.html#E2:95_4_1"><i><font color="Green" title="E81">A23</font></i></a>, <a class="txt" href="turing_1.html#E11:95_4_1"><i><font color="Green" title="E86">A28</font></i></a>, <a class="txt" href="turing_1.html#E13:95_4_1"><i><font color="Green" title="E88">A30</font></i></a>, <a class="ref" href="mcart_1.html#D7" title="MCART_1:def.7">MCART_1:def 7</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:95_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c16">s2m</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">m</font>;<br/>
<b>set </b><font color="Maroon" title="c17">skm</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">m</font>;<br/>
<a NAME="E13:95"/>
for <font color="Olive" title="b1">i</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>2</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="turing_1.html#E11:95"><i><font color="Green" title="E77">A19</font></i></a>, <a class="txt" href="turing_1.html#E12:95"><i><font color="Green" title="E78">A20</font></i></a>);<br/></i>
<a NAME="E14:95"/><b>then </b><i><font color="Green" title="E79">A31</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <font color="Maroon" title="c1">tm1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">m</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  )
 
;<br/>
<a NAME="E15:95"/><i><font color="Green" title="E80">A32</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">m</font>
 
<b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self" title="TURING_1:th.13">Th13</a></i>;<br/>
<a NAME="E16:95"/><b>then </b><i><font color="Green" title="E81">A33</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="turing_1.html#U5" title="TURING_1:sel.5">AcceptS</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">tm1</font> <a href="turing_1.html#K32" title="TURING_1:func.32">';'</a> <font color="Maroon" title="c2">tm2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="turing_1.html#E10:95"><i><font color="Green" title="E76">A18</font></i></a>, <a class="txt" href="turing_1.html#E14:95"><i><font color="Green" title="E79">A31</font></i></a>, <a class="ref" href="turing_1.html#D32" target="_self" title="TURING_1:def.32">Def32</a></i>;<br/>
<b>hence </b><a NAME="E17:95"/>
<font color="Maroon" title="c7">s3</font> is <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>
 <b>by </b><i><a class="ref" href="turing_1.html#D9" target="_self" title="TURING_1:def.9">Def9</a></i>;<br/>

<b>hence </b><a NAME="E18:95"/>
( <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c7">s3</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> <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c7">s3</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  )
 <b>by </b><i><a class="txt" href="turing_1.html#E10:95"><i><font color="Green" title="E76">A18</font></i></a>, <a class="txt" href="turing_1.html#E14:95"><i><font color="Green" title="E79">A31</font></i></a>, <a class="txt" href="turing_1.html#E15:95"><i><font color="Green" title="E80">A32</font></i></a>, <a class="txt" href="turing_1.html#E16:95"><i><font color="Green" title="E81">A33</font></i></a>, <a class="ref" href="turing_1.html#D10" target="_self" title="TURING_1:def.10">Def10</a></i>;<br/>


</div>
