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<div class="add">

<b>let </b><font color="Maroon" title="c1">tm</font> be    <a href="turing_1.html#L1" title="TURING_1:struct.1">TuringStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">s</font> be   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <font color="Maroon" title="c1">tm</font>;<br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <font color="Maroon" title="c1">tm</font>;<br/><b>let </b><font color="Maroon" title="c4">h</font> be    <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>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">tm</font>;<br/><b>let </b><font color="Maroon" title="c6">m</font>, <font color="Maroon" title="c7">d</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="E1:49"/><i><font color="Green" title="E36">A1</font></i>: 
( <font color="Maroon" title="c7">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> &amp; 1 is   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c1">tm</font> &amp; <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</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; the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c1">tm</font> <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"><font color="Maroon" title="c3">p</font>,1</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#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,1,1</span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Maroon" title="c3">p</font> <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">tm</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">m</font> holds <br/><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 ;<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="c6">m</font> implies <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <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"><font color="Maroon" title="c3">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> );<br/>
<a NAME="E2:49"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="turing_1.html#E1:49"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>;<br/>
<a NAME="E3:49"/><i><font color="Green" title="E38">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:49_1"/><i><font color="Green" title="E39">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1"/><i><font color="Green" title="E40">A5</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 ;<br/><a NAME="E2:49_1_1"/><b>then </b><i><font color="Green" title="E41">A6</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>set </b><font color="Maroon" title="c9">dk</font> = <font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font>;<br/><b>reconsider </b><font color="Maroon" title="c10">ik</font> = <font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font> 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>  <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/><b>set </b><font color="Maroon" title="c11">sk</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c10">ik</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>;<br/><b>reconsider </b><font color="Maroon" title="c12">tt</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c10">ik</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></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">tm</font> ;<br/><a NAME="E5:49_1_1"/>
<font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="txt" href="turing_1.html#E2:49_1_1"><i><font color="Green" title="E41">A6</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E6:49_1_1"/><b>then </b><i><font color="Green" title="E42">A7</font></i>: 
<font color="Maroon" title="c5">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">ik</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="turing_1.html#E1:49"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><i><font color="Green" title="E43">A8</font></i>:  <a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c10">ik</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c1">tm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">tt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c10">ik</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>
<br/>.= 
the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <font color="Maroon" title="c1">tm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c10">ik</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,1,1</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#E1:49"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="turing_1.html#E6:49_1_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c13">T</font> = 1 as   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <font color="Maroon" title="c1">tm</font> <b>by </b><i><a class="txt" href="turing_1.html#E1:49"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/><i><font color="Green" title="E44">A9</font></i>:  <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c10">ik</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c10">ik</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c10">ik</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> = 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c5">t</font>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c10">ik</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c10">ik</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c5">t</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c10">ik</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c5">t</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">T</font>
<b>by </b><i><a class="txt" href="turing_1.html#E7:49_1_1"><i><font color="Green" title="E43">A8</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="c5">t</font>
<b>by </b><i><a class="txt" href="turing_1.html#E6:49_1_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="turing_1.html#T28" target="_self" title="TURING_1:th.28">Th28</a></i>
;<br/><a NAME="E10:49_1_1"/><i><font color="Green" title="E45">A10</font></i>: 
 <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"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c10">ik</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="turing_1.html#E7:49_1_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
 <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c10">ik</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#E1:49_1"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="turing_1.html#E1:49_1_1"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c10">ik</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></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"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c10">ik</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></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"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c10">ik</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c5">t</font></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#E1:49"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="turing_1.html#E9:49_1_1"><i><font color="Green" title="E44">A9</font></i></a>, <a class="ref" href="turing_1.html#T29" target="_self" title="TURING_1:th.29">Th29</a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<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"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c10">ik</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></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"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c10">ik</font>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c5">t</font></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#E7:49_1_1"><i><font color="Green" title="E43">A8</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"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<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#E10:49_1_1"><i><font color="Green" title="E45">A10</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"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:49_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:49"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<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#E2:49"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="turing_1.html#E3:49"><i><font color="Green" title="E38">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:49"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">m</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"><font color="Maroon" title="c3">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 ;<br/>


</div>
