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

<b>let </b><font color="Maroon">tm1</font> be    <a href="turing_1.html#L1" target="_self">TuringStr</a> , <font color="Maroon">tm2</font> be    <a href="turing_1.html#L1" target="_self">TuringStr</a> ;<br/><b>let </b><font color="Maroon">s1</font> be   <a href="turing_1.html#NM4">All-State</a> of <font color="Maroon">tm1</font>;<br/><b>let </b><font color="Maroon">h</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">t</font> be   <a href="turing_1.html#NM2">Tape</a> of <font color="Maroon">tm1</font>;<br/><b>let </b><font color="Maroon">s2</font> be   <a href="turing_1.html#NM4">All-State</a> of <font color="Maroon">tm2</font>;<br/><b>let </b><font color="Maroon">s3</font> be   <a href="turing_1.html#NM4">All-State</a> of <span class="p1">(<span class="default"><font color="Maroon">tm1</font> <a href="turing_1.html#K32" target="_self">';'</a> <font color="Maroon">tm2</font></span>)</span>;<br/>











<b>set </b><font color="Maroon">p0</font> = the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">tm1</font>;<br/>
<b>set </b><font color="Maroon">q0</font> = the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">tm2</font>;<br/>
<b>assume </b><a NAME="E1:97"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">s1</font> is <a href="turing_1.html#V2" target="_self">Accept-Halt</a> &amp; <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default">the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">tm1</font>,<font color="Maroon">h</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span> &amp; <font color="Maroon">s2</font> is <a href="turing_1.html#V2" target="_self">Accept-Halt</a> &amp; <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default">the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">tm2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">s1</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">s1</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span></span><a href="domain_1.html#K4">]</a></span> &amp; <font color="Maroon">s3</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default">the <a href="turing_1.html#U4" target="_self">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon">tm1</font> <a href="turing_1.html#K32" target="_self">';'</a> <font color="Maroon">tm2</font></span>)</span>,<font color="Maroon">h</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span> )
 ;<br/>

<b>set </b><font color="Maroon">pF</font> = the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">tm1</font>;<br/>
<b>set </b><font color="Maroon">qF</font> = the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">tm2</font>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:97"/><i><font color="Green">E31</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#R1">=</a> the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">tm1</font> &amp;  <a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">tm1</font> ) )
 <b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self">Th13</a>, </i>;<br/>
<a NAME="E4:97"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">s3</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">tm1</font>,the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">tm2</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">h</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self">Th13</a>, </i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> implies ( <span class="p1"><a href="domain_1.html#K1">[</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" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">tm2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a>  ) );<br/>
<a NAME="E5:97"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<i><font color="Green">E35</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K5">`1</a>  = 
<font color="Maroon">s1</font> <a href="mcart_1.html#K5">`1</a> 
<b>by </b><i/>
<br/>.= 
the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">tm1</font>
<b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self">Th13</a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K5">`1</a>  = 
<font color="Maroon">s3</font> <a href="mcart_1.html#K5">`1</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">tm1</font>,the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">tm2</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E6:97"><i><font color="Green">E75</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/>
<b>hence </b><a NAME="E4:97_1"/>
<span class="p1"><a href="domain_1.html#K1">[</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" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">tm2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K5">`1</a> 
 <b>by </b><i><a class="ref" href="turing_1.html#D9" target="_self">Def9</a></i>;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K6">`2</a>  = 
<font color="Maroon">s1</font> <a href="mcart_1.html#K6">`2</a> 
<b>by </b><i/>
<br/>.= 
<font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self">Th13</a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<font color="Maroon">s3</font> <a href="mcart_1.html#K6">`2</a> 
<b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self">Th13</a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K6">`2</a> 
<b>by </b><i/>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K7">`3</a>  = 
<font color="Maroon">s1</font> <a href="mcart_1.html#K7">`3</a> 
<b>by </b><i/>
<br/>.= 
<font color="Maroon">t</font>
<b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self">Th13</a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<font color="Maroon">s3</font> <a href="mcart_1.html#K7">`3</a> 
<b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self">Th13</a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K7">`3</a> 
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E6:97"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:97_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_2_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><b>set </b><font color="Maroon">s1i1</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">s1i</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>;<br/><b>set </b><font color="Maroon">s3i1</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">s3i</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>;<br/><a NAME="E2:97_2_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E3:97_2_1"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>reconsider </b><font color="Maroon">ss1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K7">`3</a>  as   <a href="turing_1.html#NM2">Tape</a> of <font color="Maroon">tm1</font> ;<br/><b>reconsider </b><font color="Maroon">ss3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K7">`3</a>  as   <a href="turing_1.html#NM2">Tape</a> of <span class="p1">(<span class="default"><font color="Maroon">tm1</font> <a href="turing_1.html#K32" target="_self">';'</a> <font color="Maroon">tm2</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">h</font> =  <a href="turing_1.html#K7" target="_self">Head</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  ;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">ss1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font> as   <a href="turing_1.html#NM3">Symbol</a> of <font color="Maroon">tm1</font> ;<br/><b>set </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K5">`1</a> ;<br/><b>set </b><font color="Maroon">g</font> =  <a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>;<br/><a NAME="E8:97_2_1"/><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">tm1</font>
 <b>by </b><i><a class="ref" href="turing_1.html#T14" target="_self">Th14</a>, </i>;<br/><a NAME="E9:97_2_1"/><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <span class="p1">(<span class="default"><font color="Maroon">tm1</font> <a href="turing_1.html#K32" target="_self">';'</a> <font color="Maroon">tm2</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#R1">=</a> the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <span class="p1">(<span class="default"><font color="Maroon">tm1</font> <a href="turing_1.html#K32" target="_self">';'</a> <font color="Maroon">tm2</font></span>)</span>
 ;<br/>

<a NAME="E2:97_2_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</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" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">tm2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">tm1</font>,the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">tm2</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="turing_1.html#D10" target="_self">Def10</a>, , , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E3:97_2_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:97_2_1"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="turing_1.html#K7" target="_self">Head</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="turing_1.html#D10" target="_self">Def10</a>, , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>set </b><font color="Maroon">f</font> =  <a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>;<br/><i><font color="Green">E125</font></i>:  <a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> = 
the <a href="turing_1.html#U3" target="_self">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon">tm1</font> <a href="turing_1.html#K32" target="_self">';'</a> <font color="Maroon">tm2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</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" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">tm2</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#D10" target="_self">Def10</a>, , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">tm2</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i>, </i>
;<br/><a NAME="E12:97_2_1"/><b>then </b><i><font color="Green">E126</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a> 
 <b>by </b><i><a class="ref" href="mcart_1.html#D6">MCART_1:def 6</a></i>;<br/><i><font color="Green">E127</font></i>: <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">ss1</font>,<font color="Maroon">h</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i>, </i>
;<br/><i><font color="Green">E129</font></i>: <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">ss3</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#T16" target="_self">Th16</a>, </i>
;<br/><b>thus </b><span class="p1"><a href="domain_1.html#K1">[</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" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">tm2</font></span><a href="domain_1.html#K1">]</a></span> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">tm2</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, <a class="ref" href="mcart_1.html#D5">MCART_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D5">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" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D5">MCART_1:def 5</a></i>
;<br/><a NAME="E16:97_2_1"/>
 <a href="turing_1.html#K6" target="_self">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="turing_1.html#K6" target="_self">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="mcart_1.html#D7">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" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a>  = 
<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="mcart_1.html#D6">MCART_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D6">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" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K7">`3</a>  = 
<font color="Maroon">ss3</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="turing_1.html#K2" target="_self">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="mcart_1.html#D7">MCART_1:def 7</a>, <a class="ref" href="turing_1.html#D10" target="_self">Def10</a>, , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K7">`3</a> 
<b>by </b><i>, , , <a class="ref" href="mcart_1.html#D7">MCART_1:def 7</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:97_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">s1k</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>;<br/>
<b>set </b><font color="Maroon">s3k</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>;<br/>
<a NAME="E7:97"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="turing_1.html#T15" target="_self">Th15</a>, <a class="ref" href="turing_1.html#D9" target="_self">Def9</a>);<br/></i>
<a NAME="E8:97"/><b>then </b><i><font color="Green">E130</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</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" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">tm2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K5">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K6">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K7">`3</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K7">`3</a>  )
 
;<br/>
<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:97"/><i><font color="Green">E131</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#R1">=</a> the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">tm2</font> &amp;  <a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">tm2</font> ) )
 <b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self">Th13</a>, </i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> implies ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">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" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a>  ) );<br/>
<a NAME="E11:97"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<b>thus </b><span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">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" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">tm1</font>,<span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="mcart_1.html#K5">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">tm1</font>,the <a href="turing_1.html#U4" target="_self">InitS</a> of <font color="Maroon">tm2</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self">Th13</a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K5">`1</a> 
<b>by </b><i><a class="ref" href="turing_1.html#T14" target="_self">Th14</a>, <a class="ref" href="turing_1.html#D10" target="_self">Def10</a>, </i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K6">`2</a>  = 
<font color="Maroon">s2</font> <a href="mcart_1.html#K6">`2</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K6">`2</a> 
<b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self">Th13</a>, <a class="ref" href="turing_1.html#T14" target="_self">Th14</a>, <a class="ref" href="turing_1.html#D10" target="_self">Def10</a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K6">`2</a> 
<b>by </b><i/>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K7">`3</a>  = 
<font color="Maroon">s2</font> <a href="mcart_1.html#K7">`3</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K7">`3</a> 
<b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self">Th13</a>, <a class="ref" href="turing_1.html#T14" target="_self">Th14</a>, <a class="ref" href="turing_1.html#D10" target="_self">Def10</a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K7">`3</a> 
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E12:97"/><i><font color="Green">E161</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:97_4"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_4_1"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><b>set </b><font color="Maroon">s2i1</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">s2i</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>;<br/><b>set </b><font color="Maroon">ski1</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">ski</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>;<br/><a NAME="E2:97_4_1"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E3:97_4_1"/><b>then </b><i><font color="Green">E165</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="turing_1.html#T16" target="_self">Th16</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>reconsider </b><font color="Maroon">ss2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K7">`3</a>  as   <a href="turing_1.html#NM2">Tape</a> of <font color="Maroon">tm2</font> ;<br/><b>reconsider </b><font color="Maroon">ssk</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K7">`3</a>  as   <a href="turing_1.html#NM2">Tape</a> of <span class="p1">(<span class="default"><font color="Maroon">tm1</font> <a href="turing_1.html#K32" target="_self">';'</a> <font color="Maroon">tm2</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">h</font> =  <a href="turing_1.html#K7" target="_self">Head</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  ;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">ss2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font> as   <a href="turing_1.html#NM3">Symbol</a> of <font color="Maroon">tm2</font> ;<br/><b>set </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K5">`1</a> ;<br/><b>set </b><font color="Maroon">g</font> =  <a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>;<br/><a NAME="E8:97_4_1"/><i><font color="Green">E166</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">tm2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E9:97_4_1"/><i><font color="Green">E170</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <span class="p1">(<span class="default"><font color="Maroon">tm1</font> <a href="turing_1.html#K32" target="_self">';'</a> <font color="Maroon">tm2</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_4_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_4_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#R1">=</a> the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <span class="p1">(<span class="default"><font color="Maroon">tm1</font> <a href="turing_1.html#K32" target="_self">';'</a> <font color="Maroon">tm2</font></span>)</span>
 ;<br/>

<a NAME="E2:97_4_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">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" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">tm1</font>,the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">tm2</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, <a class="ref" href="turing_1.html#T16" target="_self">Th16</a>, , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E3:97_4_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:97_4_1"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="turing_1.html#K7" target="_self">Head</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="turing_1.html#T16" target="_self">Th16</a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>set </b><font color="Maroon">f</font> =  <a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>;<br/><i><font color="Green">E173</font></i>:  <a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> = 
the <a href="turing_1.html#U3" target="_self">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon">tm1</font> <a href="turing_1.html#K32" target="_self">';'</a> <font color="Maroon">tm2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">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" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, <a class="ref" href="turing_1.html#T16" target="_self">Th16</a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">tm1</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i/>
;<br/><a NAME="E12:97_4_1"/><b>then </b><i><font color="Green">E174</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a> 
 <b>by </b><i><a class="ref" href="mcart_1.html#D6">MCART_1:def 6</a></i>;<br/><i><font color="Green">E175</font></i>: <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">ss2</font>,<font color="Maroon">h</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i>, </i>
;<br/><i><font color="Green">E176</font></i>: <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">ssk</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i>, </i>
;<br/><b>thus </b><span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">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" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">tm1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, <a class="ref" href="mcart_1.html#D5">MCART_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D5">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" target="_self">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D5">MCART_1:def 5</a></i>
;<br/><a NAME="E16:97_4_1"/>
 <a href="turing_1.html#K6" target="_self">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="turing_1.html#K6" target="_self">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="mcart_1.html#D7">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" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a>  = 
<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="mcart_1.html#D6">MCART_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D6">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" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K7">`3</a>  = 
<font color="Maroon">ssk</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="turing_1.html#K2" target="_self">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="mcart_1.html#D7">MCART_1:def 7</a>, , <a class="ref" href="turing_1.html#T16" target="_self">Th16</a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K7">`3</a> 
<b>by </b><i>, , , <a class="ref" href="mcart_1.html#D7">MCART_1:def 7</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:97_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">s2m</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>;<br/>
<b>set </b><font color="Maroon">skm</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>;<br/>
<a NAME="E13:97"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E14:97"/><b>then </b><i><font color="Green">E177</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">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" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K5">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K6">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K7">`3</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K7">`3</a>  )
 
;<br/>
<a NAME="E15:97"/><i><font color="Green">E178</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<a NAME="E16:97"/><b>then </b><i><font color="Green">E179</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#R1">=</a> the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <span class="p1">(<span class="default"><font color="Maroon">tm1</font> <a href="turing_1.html#K32" target="_self">';'</a> <font color="Maroon">tm2</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E17:97"/>
<font color="Maroon">s3</font> is <a href="turing_1.html#V2" target="_self">Accept-Halt</a>
 <b>by </b><i/>;<br/>

<b>hence </b><a NAME="E18:97"/>
( <span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">s3</font></span>)</span> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">s2</font></span>)</span> <a href="mcart_1.html#K6">`2</a>  &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">s3</font></span>)</span> <a href="mcart_1.html#K7">`3</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">s2</font></span>)</span> <a href="mcart_1.html#K7">`3</a>  )
 <b>by </b><i>, , <a class="ref" href="turing_1.html#T16" target="_self">Th16</a>, , </i>;<br/>


</div>
