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<b>let </b><font color="Maroon" title="c1">s</font> be   <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a> ;<br/><b>let </b><font color="Maroon" title="c2">t</font> be   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a> ;<br/><b>let </b><font color="Maroon" title="c3">h</font>, <font color="Maroon" title="c4">n1</font>, <font color="Maroon" title="c5">n2</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="E1:50"/><i><font color="Green" title="E40">A1</font></i>: 
( <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default">0,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Maroon" title="c2">t</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c4">n1</font>,<font color="Maroon" title="c5">n2</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c6">s3</font> = <font color="Maroon" title="c1">s</font> <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 <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">F</font> = 0 as   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E48"><i><font color="Green" title="E34">Lm5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">T</font> = 1 as   <a href="turing_1.html#NM3" title="TURING_1:NM.3">Symbol</a> of <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E48"><i><font color="Green" title="E34">Lm5</font></i></a></i>;<br/>
<i><font color="Green" title="E41">A2</font></i>:  <a href="turing_1.html#K8" title="TURING_1:func.8">TRAN</a> <font color="Maroon" title="c1">s</font> = 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">s3</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> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#D15" target="_self" title="TURING_1:def.15">Def15</a></i>
<br/>.= 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c6">s3</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> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:50"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>
<br/>.= 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c2">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> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:50"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>
<br/>.= 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c2">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">h</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:50"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default">0,0,1</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#E1:50"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="turing_1.html#T17" target="_self" title="TURING_1:th.17">Th17</a>, <a class="ref" href="turing_1.html#T22" target="_self" title="TURING_1:th.22">Th22</a></i>
;<br/>
<a NAME="E6:50"/><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon" title="c2">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="turing_1.html#E1:50"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="turing_1.html#T22" target="_self" title="TURING_1:th.22">Th22</a></i>;<br/>
<i><font color="Green" title="E43">A4</font></i>:  <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <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> <font color="Maroon" title="c1">s</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> <font color="Maroon" title="c1">s</font></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="c2">t</font>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" title="TURING_1:func.7">Head</a> <font color="Maroon" title="c1">s</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> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:50"><i><font color="Green" title="E40">A1</font></i></a>, <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="c2">t</font>,<font color="Maroon" title="c3">h</font>,<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> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:50"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c7">F</font>
<b>by </b><i><a class="txt" href="turing_1.html#E5:50"><i><font color="Green" title="E41">A2</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="c2">t</font>
<b>by </b><i><a class="txt" href="turing_1.html#E6:50"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="turing_1.html#T28" target="_self" title="TURING_1:th.28">Th28</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c9">p0</font> = 0 as   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E44"><i><font color="Green" title="E31">Lm3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">h1</font> = <font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 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/>
<a NAME="E10:50"/><i><font color="Green" title="E44">A5</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> <font color="Maroon" title="c1">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="turing_1.html#E5:50"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">s1</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>;<br/>
<i><font color="Green" title="E45">A6</font></i>:  <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <font color="Maroon" title="c1">s</font> = 
<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> <font color="Maroon" title="c1">s</font></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> <font color="Maroon" title="c1">s</font></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> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">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#E1:50"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="turing_1.html#E7:50"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="turing_1.html#E50"><i><font color="Green" title="E36">Lm6</font></i></a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default">0,<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> <font color="Maroon" title="c1">s</font></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> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">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#E5:50"><i><font color="Green" title="E41">A2</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="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:50"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="turing_1.html#E10:50"><i><font color="Green" title="E44">A5</font></i></a>, <a class="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c12">s3</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">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 <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>  ;<br/>
<a NAME="E13:50"/><i><font color="Green" title="E46">A7</font></i>: 
( <font color="Maroon" title="c2">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c2">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c2">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &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="c3">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 holds <br/><font color="Maroon" title="c2">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 ) &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 holds <br/><font color="Maroon" title="c2">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 ) )
 
<b>by </b><i><a class="txt" href="turing_1.html#E1:50"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="turing_1.html#T22" target="_self" title="TURING_1:th.22">Th22</a></i>;<br/>
<a NAME="E14:50"/><i><font color="Green" title="E47">A8</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">h1</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E15:50"/>
<font color="Maroon" title="c3">h</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E16:50"/><b>then </b><i><font color="Green" title="E48">A9</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E17:50"/><i><font color="Green" title="E49">A10</font></i>: 
<font color="Maroon" title="c10">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E18:50"/><b>then </b><i><font color="Green" title="E50">A11</font></i>: 
<font color="Maroon" title="c10">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<b>by </b><i><a class="txt" href="turing_1.html#E16:50"><i><font color="Green" title="E48">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:50"/><i><font color="Green" title="E51">A12</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
;<br/>
<b>set </b><font color="Maroon" title="c13">t1</font> =  <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font>;<br/>
<a NAME="E20:50"/>
<font color="Maroon" title="c3">h</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E21:50"/><b>then </b><i><font color="Green" title="E52">A13</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E22:50"/><i><font color="Green" title="E53">A14</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th30</a></i>;<br/>
<a NAME="E23:50"/><i><font color="Green" title="E54">A15</font></i>: 
( <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &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="c10">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font> holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span> <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 ) &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span> <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 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:50_4"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="turing_1.html#E13:50"><i><font color="Green" title="E46">A7</font></i></a>, <a class="txt" href="turing_1.html#E14:50"><i><font color="Green" title="E47">A8</font></i></a>, <a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th30</a></i>;<br/>

<b>thus </b><a NAME="E2:50_4"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="turing_1.html#E13:50"><i><font color="Green" title="E46">A7</font></i></a>, <a class="txt" href="turing_1.html#E16:50"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="turing_1.html#E17:50"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th30</a></i>;<br/>

<a NAME="E3:50_4"/>
<font color="Maroon" title="c10">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E4:50_4"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="turing_1.html#E13:50"><i><font color="Green" title="E46">A7</font></i></a>, <a class="txt" href="turing_1.html#E21:50"><i><font color="Green" title="E52">A13</font></i></a>, <a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th30</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c14">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><b>assume </b><a NAME="E1:50_4_1"/><i><font color="Green" title="E55">A16</font></i>: 
( <font color="Maroon" title="c10">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font> )
 ;<br/><a NAME="E2:50_4_1"/><b>then </b><i><font color="Green" title="E56">A17</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="turing_1.html#E14:50"><i><font color="Green" title="E47">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> = 
<font color="Maroon" title="c2">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
<b>by </b><i><a class="txt" href="turing_1.html#E1:50_4_1"><i><font color="Green" title="E55">A16</font></i></a>, <a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th30</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="turing_1.html#E1:50"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="turing_1.html#E19:50"><i><font color="Green" title="E51">A12</font></i></a>, <a class="txt" href="turing_1.html#E1:50_4_1"><i><font color="Green" title="E55">A16</font></i></a>, <a class="txt" href="turing_1.html#E2:50_4_1"><i><font color="Green" title="E56">A17</font></i></a>, <a class="ref" href="turing_1.html#T22" target="_self" title="TURING_1:th.22">Th22</a></i>
;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c14">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><b>assume </b><a NAME="E1:50_4_2"/><i><font color="Green" title="E55">A18</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 )
 ;<br/><b>hence </b><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> = 
<font color="Maroon" title="c2">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
<b>by </b><i><a class="txt" href="turing_1.html#E18:50"><i><font color="Green" title="E50">A11</font></i></a>, <a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th30</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="turing_1.html#E1:50"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="turing_1.html#E1:50_4_2"><i><font color="Green" title="E55">A18</font></i></a>, <a class="ref" href="turing_1.html#T22" target="_self" title="TURING_1:th.22">Th22</a></i>
;<br/><br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<i><font color="Green" title="E55">A19</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="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">s3</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="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">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="turing_1.html#D15" target="_self" title="TURING_1:def.15">Def15</a></i>
<br/>.= 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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="c9">p0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">s3</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="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">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/>.= 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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="c9">p0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">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="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">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/>.= 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c2">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h1</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default">1,0,1</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#E1:50"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="turing_1.html#E14:50"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="turing_1.html#E18:50"><i><font color="Green" title="E50">A11</font></i></a>, <a class="ref" href="turing_1.html#T17" target="_self" title="TURING_1:th.17">Th17</a>, <a class="ref" href="turing_1.html#T22" target="_self" title="TURING_1:th.22">Th22</a></i>
;<br/>
<i><font color="Green" title="E56">A20</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="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">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="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">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="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">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="c2">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="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">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="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">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="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<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="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">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="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font>
<b>by </b><i><a class="txt" href="turing_1.html#E24:50"><i><font color="Green" title="E55">A19</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>
;<br/>
<a NAME="E26:50"/><i><font color="Green" title="E57">A21</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="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">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#E24:50"><i><font color="Green" title="E55">A19</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="c14">p1</font> = 1 as   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E44"><i><font color="Green" title="E31">Lm3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">i2</font> = <font color="Maroon" title="c10">h1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 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="c16">s2</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c15">i2</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>;<br/>
<i><font color="Green" title="E58">A22</font></i>:  <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="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<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="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">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="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">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="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E25:50"><i><font color="Green" title="E56">A20</font></i></a>, <a class="txt" href="turing_1.html#E50"><i><font color="Green" title="E36">Lm6</font></i></a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default">1,<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="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">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="c9">p0</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c2">t</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E24:50"><i><font color="Green" title="E55">A19</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="c14">p1</font>,<font color="Maroon" title="c15">i2</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E26:50"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
;<br/>
<a NAME="E30:50"/><i><font color="Green" title="E59">A23</font></i>: 
the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>  <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="c14">p1</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="c14">p1</font>,1,1</span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="turing_1.html#D15" target="_self" title="TURING_1:def.15">Def15</a>, <a class="ref" href="turing_1.html#T17" target="_self" title="TURING_1:th.17">Th17</a></i>;<br/>
<a NAME="E31:50"/><i><font color="Green" title="E60">A24</font></i>: 
<font color="Maroon" title="c14">p1</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 <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a> 
 
<b>by </b><i><a class="ref" href="turing_1.html#D15" target="_self" title="TURING_1:def.15">Def15</a></i>;<br/>
<a NAME="E32:50"/><i><font color="Green" title="E61">A25</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font> holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span> <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
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<b>assume </b><a NAME="E1:50_8"/><i><font color="Green" title="E62">A26</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font> &amp; <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font> )
 ;<br/>

<a NAME="E2:50_8"/>
<font color="Maroon" title="c10">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">h1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E3:50_8"/><b>then </b>
<font color="Maroon" title="c10">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">i</font>
 
<b>by </b><i><a class="txt" href="turing_1.html#E1:50_8"><i><font color="Green" title="E62">A26</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E4:50_8"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="turing_1.html#E23:50"><i><font color="Green" title="E54">A15</font></i></a>, <a class="txt" href="turing_1.html#E1:50_8"><i><font color="Green" title="E62">A26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c17">nk</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c10">h1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">n1</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="c18">sn</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c17">nk</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">sn3</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c17">nk</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></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 <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>  ;<br/>
<div><i><font color="Green" title="E62">A27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_9"><b>now </b></a><div class="add"><b>thus </b> <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="c14">p1</font>,<font color="Maroon" title="c17">nk</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c17">nk</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">sn3</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="c14">p1</font>,<font color="Maroon" title="c17">nk</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></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="turing_1.html#D15" target="_self" title="TURING_1:def.15">Def15</a></i>
<br/>.= 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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="c14">p1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">sn3</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="c14">p1</font>,<font color="Maroon" title="c17">nk</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></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/>.= 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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="c14">p1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span> <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="c14">p1</font>,<font color="Maroon" title="c17">nk</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></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">2,1,1</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#E23:50"><i><font color="Green" title="E54">A15</font></i></a>, <a class="ref" href="turing_1.html#T17" target="_self" title="TURING_1:th.17">Th17</a>, <a class="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c20">t2</font> =  <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font>;<br/>
<a NAME="E36:50"/><i><font color="Green" title="E63">A28</font></i>: 
<font color="Maroon" title="c10">h1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E37:50"/><i><font color="Green" title="E64">A29</font></i>: 
<font color="Maroon" title="c10">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">h1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E38:50"/><b>then </b><i><font color="Green" title="E65">A30</font></i>: 
<font color="Maroon" title="c10">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">n1</font>
 
<b>by </b><i><a class="txt" href="turing_1.html#E36:50"><i><font color="Green" title="E63">A28</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E39:50"/><i><font color="Green" title="E66">A31</font></i>: 
( <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &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="c10">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span> <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 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_10"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:50_10"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="turing_1.html#E22:50"><i><font color="Green" title="E53">A14</font></i></a>, <a class="txt" href="turing_1.html#E36:50"><i><font color="Green" title="E63">A28</font></i></a>, <a class="txt" href="turing_1.html#E37:50"><i><font color="Green" title="E64">A29</font></i></a>, <a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th30</a></i>;<br/>

<a NAME="E2:50_10"/>
<font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E3:50_10"/><b>then </b><i><font color="Green" title="E67">A32</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<b>by </b><i><a class="txt" href="turing_1.html#E19:50"><i><font color="Green" title="E51">A12</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E4:50_10"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E5:50_10"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="turing_1.html#E23:50"><i><font color="Green" title="E54">A15</font></i></a>, <a class="txt" href="turing_1.html#E3:50_10"><i><font color="Green" title="E67">A32</font></i></a>, <a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th30</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c21">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><b>assume </b><a NAME="E1:50_10_1"/><i><font color="Green" title="E68">A33</font></i>: 
( <font color="Maroon" title="c10">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_10_1_1"/>
( <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font> or <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font> or <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_10_1_1_1"><b>suppose </b></a><a NAME="E1:50_10_1_1_1"/><i><font color="Green" title="E69">A34</font></i>: 
<font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> = 
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
<b>by </b><i><a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th30</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="turing_1.html#E23:50"><i><font color="Green" title="E54">A15</font></i></a>, <a class="txt" href="turing_1.html#E1:50_10_1"><i><font color="Green" title="E68">A33</font></i></a>, <a class="txt" href="turing_1.html#E1:50_10_1_1_1"><i><font color="Green" title="E69">A34</font></i></a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_10_1_1_2"><b>suppose </b></a><a NAME="E1:50_10_1_1_2"/>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:50_10_1_1_2"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th30</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_10_1_1_3"><b>suppose </b></a><a NAME="E1:50_10_1_1_3"/><i><font color="Green" title="E69">A35</font></i>: 
<font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> = 
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
<b>by </b><i><a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th30</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="turing_1.html#E23:50"><i><font color="Green" title="E54">A15</font></i></a>, <a class="txt" href="turing_1.html#E1:50_10_1"><i><font color="Green" title="E68">A33</font></i></a>, <a class="txt" href="turing_1.html#E1:50_10_1_1_3"><i><font color="Green" title="E69">A35</font></i></a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>


</div><b>end;</b></div>
<i><font color="Green" title="E67">A36</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="c14">p1</font>,<font color="Maroon" title="c17">nk</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></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="c14">p1</font>,<font color="Maroon" title="c17">nk</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></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="c14">p1</font>,<font color="Maroon" title="c17">nk</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></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> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></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="c14">p1</font>,<font color="Maroon" title="c17">nk</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></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="c14">p1</font>,<font color="Maroon" title="c17">nk</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></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> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<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="c14">p1</font>,<font color="Maroon" title="c17">nk</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></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="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font>
<b>by </b><i><a class="txt" href="turing_1.html#E35:50"><i><font color="Green" title="E62">A27</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>
;<br/>
<a NAME="E41:50"/><i><font color="Green" title="E68">A37</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="c14">p1</font>,<font color="Maroon" title="c17">nk</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></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#E35:50"><i><font color="Green" title="E62">A27</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>;<br/>
<b>set </b><font color="Maroon" title="c21">i3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<b>reconsider </b><font color="Maroon" title="c22">n3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 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>reconsider </b><font color="Maroon" title="c23">p2</font> = 2 as   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E44"><i><font color="Green" title="E31">Lm3</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c24">sm</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c23">p2</font>,<font color="Maroon" title="c22">n3</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>;<br/>
<i><font color="Green" title="E69">A38</font></i>:  <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="c14">p1</font>,<font color="Maroon" title="c17">nk</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<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="c14">p1</font>,<font color="Maroon" title="c17">nk</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></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="c14">p1</font>,<font color="Maroon" title="c17">nk</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></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="c14">p1</font>,<font color="Maroon" title="c17">nk</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E40:50"><i><font color="Green" title="E67">A36</font></i></a>, <a class="txt" href="turing_1.html#E50"><i><font color="Green" title="E36">Lm6</font></i></a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default">2,<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="c14">p1</font>,<font color="Maroon" title="c17">nk</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></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="c14">p1</font>,<font color="Maroon" title="c17">nk</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E35:50"><i><font color="Green" title="E62">A27</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="c23">p2</font>,<font color="Maroon" title="c22">n3</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E41:50"><i><font color="Green" title="E68">A37</font></i></a>, <a class="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
;<br/>
<a NAME="E45:50"/><i><font color="Green" title="E70">A39</font></i>: 
the <a href="turing_1.html#U3" title="TURING_1:sel.3">Tran</a> of <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>  <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="c23">p2</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="c23">p2</font>,1,1</span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="turing_1.html#D15" target="_self" title="TURING_1:def.15">Def15</a>, <a class="ref" href="turing_1.html#T17" target="_self" title="TURING_1:th.17">Th17</a></i>;<br/>
<a NAME="E46:50"/><i><font color="Green" title="E71">A40</font></i>: 
<font color="Maroon" title="c23">p2</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 <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a> 
 
<b>by </b><i><a class="ref" href="turing_1.html#D15" target="_self" title="TURING_1:def.15">Def15</a></i>;<br/>
<a NAME="E47:50"/>
for <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span> <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
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c25">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<b>assume </b><a NAME="E1:50_13"/><i><font color="Green" title="E72">A41</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">i</font> &amp; <font color="Maroon" title="c25">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 ;<br/>

<a NAME="E2:50_13"/>
<font color="Maroon" title="c17">nk</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E3:50_13"/><b>then </b>
<font color="Maroon" title="c10">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="turing_1.html#E38:50"><i><font color="Green" title="E65">A30</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:50_13"/><b>then </b>
<font color="Maroon" title="c10">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">i</font>
 
<b>by </b><i><a class="txt" href="turing_1.html#E1:50_13"><i><font color="Green" title="E72">A41</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:50_13"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="turing_1.html#E39:50"><i><font color="Green" title="E66">A31</font></i></a>, <a class="txt" href="turing_1.html#E1:50_13"><i><font color="Green" title="E72">A41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E48:50"/><b>then </b><i><font color="Green" title="E72">A42</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c23">p2</font>,<font color="Maroon" title="c22">n3</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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">2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="turing_1.html#E45:50"><i><font color="Green" title="E70">A39</font></i></a>, <a class="txt" href="turing_1.html#E46:50"><i><font color="Green" title="E71">A40</font></i></a>, <a class="txt" href="turing_1.html#E52"><i><font color="Green" title="E38">Lm7</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c25">i4</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4;<br/>
<b>reconsider </b><font color="Maroon" title="c26">n4</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 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="c27">sp2</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c23">p2</font>,<font color="Maroon" title="c26">n4</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c28">sn3</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c23">p2</font>,<font color="Maroon" title="c26">n4</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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 <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>  ;<br/>
<i><font color="Green" title="E73">A43</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="c23">p2</font>,<font color="Maroon" title="c26">n4</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c23">p2</font>,<font color="Maroon" title="c26">n4</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">sn3</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="c23">p2</font>,<font color="Maroon" title="c26">n4</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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="turing_1.html#D15" target="_self" title="TURING_1:def.15">Def15</a></i>
<br/>.= 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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="c23">p2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">sn3</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="c23">p2</font>,<font color="Maroon" title="c26">n4</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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/>.= 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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="c23">p2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span> <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="c23">p2</font>,<font color="Maroon" title="c26">n4</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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">3,0,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E39:50"><i><font color="Green" title="E66">A31</font></i></a>, <a class="ref" href="turing_1.html#T17" target="_self" title="TURING_1:th.17">Th17</a>, <a class="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
;<br/>
<b>set </b><font color="Maroon" title="c29">j3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1;<br/>
<div><i><font color="Green" title="E74">A44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_15"><b>now </b></a><div class="add"><i><font color="Green" title="E75">A45</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="c23">p2</font>,<font color="Maroon" title="c26">n4</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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="c23">p2</font>,<font color="Maroon" title="c26">n4</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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="c23">p2</font>,<font color="Maroon" title="c26">n4</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></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="c23">p2</font>,<font color="Maroon" title="c26">n4</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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="c23">p2</font>,<font color="Maroon" title="c26">n4</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</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="c23">p2</font>,<font color="Maroon" title="c26">n4</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span>,<font color="Maroon" title="c7">F</font>
<b>by </b><i><a class="txt" href="turing_1.html#E51:50"><i><font color="Green" title="E73">A43</font></i></a>, <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> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font>
<b>by </b><i><a class="txt" href="turing_1.html#E39:50"><i><font color="Green" title="E66">A31</font></i></a>, <a class="ref" href="turing_1.html#T28" target="_self" title="TURING_1:th.28">Th28</a></i>
;<br/><a NAME="E2:50_15"/><i><font color="Green" title="E76">A46</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="c23">p2</font>,<font color="Maroon" title="c26">n4</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="txt" href="turing_1.html#E51:50"><i><font color="Green" title="E73">A43</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>;<br/><a NAME="E3:50_15"/><i><font color="Green" title="E77">A47</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 ;<br/><b>thus </b> <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="c23">p2</font>,<font color="Maroon" title="c26">n4</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<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="c23">p2</font>,<font color="Maroon" title="c26">n4</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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="c23">p2</font>,<font color="Maroon" title="c26">n4</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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="c23">p2</font>,<font color="Maroon" title="c26">n4</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:50_15"><i><font color="Green" title="E75">A45</font></i></a>, <a class="txt" href="turing_1.html#E50"><i><font color="Green" title="E36">Lm6</font></i></a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default">3,<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="c23">p2</font>,<font color="Maroon" title="c26">n4</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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="c23">p2</font>,<font color="Maroon" title="c26">n4</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E51:50"><i><font color="Green" title="E73">A43</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">3,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E2:50_15"><i><font color="Green" title="E76">A46</font></i></a>, <a class="txt" href="turing_1.html#E3:50_15"><i><font color="Green" title="E77">A47</font></i></a>, <a class="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c30">p3</font> = 3 as   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E44"><i><font color="Green" title="E31">Lm3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c31">m3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 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="c32">sp3</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c33">sn3</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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 <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>  ;<br/>
<a NAME="E56:50"/><i><font color="Green" title="E75">A48</font></i>: 
<font color="Maroon" title="c10">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E57:50"/><i><font color="Green" title="E76">A49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<b>by </b><i><a class="txt" href="turing_1.html#E21:50"><i><font color="Green" title="E52">A13</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E58:50"/><b>then </b><i><font color="Green" title="E77">A50</font></i>: 
<font color="Maroon" title="c10">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<b>by </b><i><a class="txt" href="turing_1.html#E56:50"><i><font color="Green" title="E75">A48</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E59:50"/><i><font color="Green" title="E78">A51</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4
 
<b>by </b><i><a class="ref" href="xreal_1.html#T148" title="XREAL_1:th.148">XREAL_1:148</a></i>;<br/>
<a NAME="E60:50"/><b>then </b><i><font color="Green" title="E79">A52</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="turing_1.html#E39:50"><i><font color="Green" title="E66">A31</font></i></a>, <a class="txt" href="turing_1.html#E58:50"><i><font color="Green" title="E77">A50</font></i></a></i>;<br/>
<div><i><font color="Green" title="E80">A53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_16"><b>now </b></a><div class="add"><b>thus </b> <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="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c33">sn3</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="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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="turing_1.html#D15" target="_self" title="TURING_1:def.15">Def15</a></i>
<br/>.= 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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="c30">p3</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c33">sn3</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="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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/>.= 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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="c30">p3</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span> <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="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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">4,0,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E60:50"><i><font color="Green" title="E79">A52</font></i></a>, <a class="ref" href="turing_1.html#T17" target="_self" title="TURING_1:th.17">Th17</a>, <a class="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c34">t3</font> =  <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font>;<br/>
<a NAME="E62:50"/><i><font color="Green" title="E81">A54</font></i>: 
( <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &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="c10">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span> <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 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_17"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:50_17"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="turing_1.html#E39:50"><i><font color="Green" title="E66">A31</font></i></a>, <a class="txt" href="turing_1.html#E56:50"><i><font color="Green" title="E75">A48</font></i></a>, <a class="txt" href="turing_1.html#E57:50"><i><font color="Green" title="E76">A49</font></i></a>, <a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th30</a></i>;<br/>

<b>thus </b><a NAME="E2:50_17"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th30</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c35">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><b>assume </b><a NAME="E1:50_17_1"/><i><font color="Green" title="E82">A55</font></i>: 
( <font color="Maroon" title="c10">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">i</font> &amp; <font color="Maroon" title="c35">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 )
 ;<br/><a NAME="E2:50_17_1"/><b>then </b><i><font color="Green" title="E83">A56</font></i>: 
<font color="Maroon" title="c35">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4
 <b>by </b><i><a class="txt" href="turing_1.html#E59:50"><i><font color="Green" title="E78">A51</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font> = 
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font>
<b>by </b><i><a class="txt" href="turing_1.html#E1:50_17_1"><i><font color="Green" title="E82">A55</font></i></a>, <a class="ref" href="turing_1.html#T30" target="_self" title="TURING_1:th.30">Th30</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="turing_1.html#E39:50"><i><font color="Green" title="E66">A31</font></i></a>, <a class="txt" href="turing_1.html#E1:50_17_1"><i><font color="Green" title="E82">A55</font></i></a>, <a class="txt" href="turing_1.html#E2:50_17_1"><i><font color="Green" title="E83">A56</font></i></a></i>
;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<i><font color="Green" title="E82">A57</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="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></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="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</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="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font>
<b>by </b><i><a class="txt" href="turing_1.html#E61:50"><i><font color="Green" title="E80">A53</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>
;<br/>
<a NAME="E64:50"/><i><font color="Green" title="E83">A58</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="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 
<b>by </b><i><a class="txt" href="turing_1.html#E61:50"><i><font color="Green" title="E80">A53</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>;<br/>
<b>set </b><font color="Maroon" title="c35">j2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1;<br/>
<a NAME="E65:50"/><i><font color="Green" title="E84">A59</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
;<br/>
<b>reconsider </b><font color="Maroon" title="c36">p4</font> = 4 as   <a href="turing_1.html#NM1" title="TURING_1:NM.1">State</a> of <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E44"><i><font color="Green" title="E31">Lm3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c37">m2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 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="c38">sp4</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c36">p4</font>,<font color="Maroon" title="c37">m2</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>;<br/>
<div><i><font color="Green" title="E85">A60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_19"><b>now </b></a><div class="add"><b>thus </b> <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="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<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="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E63:50"><i><font color="Green" title="E82">A57</font></i></a>, <a class="txt" href="turing_1.html#E50"><i><font color="Green" title="E36">Lm6</font></i></a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default">4,<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="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></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="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E61:50"><i><font color="Green" title="E80">A53</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="c36">p4</font>,<font color="Maroon" title="c37">m2</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E64:50"><i><font color="Green" title="E83">A58</font></i></a>, <a class="txt" href="turing_1.html#E65:50"><i><font color="Green" title="E84">A59</font></i></a>, <a class="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
;<br/></div><b>end;</b></div>
<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>( <font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 implies <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c36">p4</font>,<font color="Maroon" title="c37">m2</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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">4,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> $1</span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> );<br/>
<a NAME="E69:50"/><i><font color="Green" title="E86">A61</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>;<br/>
<a NAME="E70:50"/><i><font color="Green" title="E87">A62</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#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_20"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c39">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="E1:50_20"/><i><font color="Green" title="E88">A63</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c39">k</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_20_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_20_1"/><i><font color="Green" title="E89">A64</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 ;<br/><a NAME="E2:50_20_1"/><i><font color="Green" title="E90">A65</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c39">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c39">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><b>set </b><font color="Maroon" title="c40">k3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c39">k</font>;<br/><b>reconsider </b><font color="Maroon" title="c41">ik</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c39">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="c42">sk</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c36">p4</font>,<font color="Maroon" title="c41">ik</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>;<br/><b>reconsider </b><font color="Maroon" title="c43">tt</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c36">p4</font>,<font color="Maroon" title="c41">ik</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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 <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>  ;<br/><a NAME="E5:50_20_1"/>
<font color="Maroon" title="c10">h1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c39">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0
 <b>by </b><i><a class="txt" href="turing_1.html#E1:50_20_1"><i><font color="Green" title="E89">A64</font></i></a></i>;<br/><a NAME="E6:50_20_1"/><b>then </b><i><font color="Green" title="E91">A66</font></i>: 
<font color="Maroon" title="c10">h1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c39">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T23" title="XREAL_1:th.23">XREAL_1:23</a></i>;<br/><a NAME="E7:50_20_1"/>
<font color="Maroon" title="c10">h1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E8:50_20_1"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c39">k</font>
 <b>by </b><i><a class="txt" href="turing_1.html#E6:50_20_1"><i><font color="Green" title="E91">A66</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c44">ii</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c39">k</font> as    <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>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E10:50_20_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c39">k</font>
 <b>by </b><i><a class="ref" href="int_1.html#T19" title="INT_1:th.19">INT_1:19</a></i>;<br/><a NAME="E11:50_20_1"/><b>then </b><i><font color="Green" title="E92">A67</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c39">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E12:50_20_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T148" title="XREAL_1:th.148">XREAL_1:148</a></i>;<br/><a NAME="E13:50_20_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c39">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="txt" href="turing_1.html#E11:50_20_1"><i><font color="Green" title="E92">A67</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E14:50_20_1"/><b>then </b><i><font color="Green" title="E93">A68</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="turing_1.html#E62:50"><i><font color="Green" title="E81">A54</font></i></a>, <a class="txt" href="turing_1.html#E6:50_20_1"><i><font color="Green" title="E91">A66</font></i></a></i>;<br/><div><i><font color="Green" title="E94">A69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_20_1_1"><b>now </b></a><div class="add"><b>thus </b> <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="c36">p4</font>,<font color="Maroon" title="c41">ik</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c36">p4</font>,<font color="Maroon" title="c41">ik</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c43">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="c36">p4</font>,<font color="Maroon" title="c41">ik</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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="turing_1.html#D15" target="_self" title="TURING_1:def.15">Def15</a></i>
<br/>.= 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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="c36">p4</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c43">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="c36">p4</font>,<font color="Maroon" title="c41">ik</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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/>.= 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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="c36">p4</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span> <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="c36">p4</font>,<font color="Maroon" title="c41">ik</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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">4,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E14:50_20_1"><i><font color="Green" title="E93">A68</font></i></a>, <a class="ref" href="turing_1.html#T17" target="_self" title="TURING_1:th.17">Th17</a>, <a class="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
;<br/></div><b>end;</b></div><div><i><font color="Green" title="E95">A70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_20_1_2"><b>now </b></a><div class="add"><b>thus </b> <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="c36">p4</font>,<font color="Maroon" title="c41">ik</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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="c36">p4</font>,<font color="Maroon" title="c41">ik</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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="c36">p4</font>,<font color="Maroon" title="c41">ik</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></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="c36">p4</font>,<font color="Maroon" title="c41">ik</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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="c36">p4</font>,<font color="Maroon" title="c41">ik</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c39">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="c36">p4</font>,<font color="Maroon" title="c41">ik</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c39">k</font></span>)</span>,<font color="Maroon" title="c8">T</font>
<b>by </b><i><a class="txt" href="turing_1.html#E15:50_20_1"><i><font color="Green" title="E94">A69</font></i></a>, <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> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font>
<b>by </b><i><a class="txt" href="turing_1.html#E14:50_20_1"><i><font color="Green" title="E93">A68</font></i></a>, <a class="ref" href="turing_1.html#T28" target="_self" title="TURING_1:th.28">Th28</a></i>
;<br/></div><b>end;</b></div><a NAME="E17:50_20_1"/><i><font color="Green" title="E96">A71</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="c36">p4</font>,<font color="Maroon" title="c41">ik</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="txt" href="turing_1.html#E15:50_20_1"><i><font color="Green" title="E94">A69</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> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c36">p4</font>,<font color="Maroon" title="c37">m2</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c36">p4</font>,<font color="Maroon" title="c41">ik</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:50_20"><i><font color="Green" title="E88">A63</font></i></a>, <a class="txt" href="turing_1.html#E1:50_20_1"><i><font color="Green" title="E89">A64</font></i></a>, <a class="txt" href="turing_1.html#E2:50_20_1"><i><font color="Green" title="E90">A65</font></i></a>, <a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
<br/>.= 
<span class="p1"><a href="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="c36">p4</font>,<font color="Maroon" title="c41">ik</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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="c36">p4</font>,<font color="Maroon" title="c41">ik</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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="c36">p4</font>,<font color="Maroon" title="c41">ik</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E16:50_20_1"><i><font color="Green" title="E95">A70</font></i></a>, <a class="txt" href="turing_1.html#E50"><i><font color="Green" title="E36">Lm6</font></i></a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default">4,<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="c36">p4</font>,<font color="Maroon" title="c41">ik</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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="c36">p4</font>,<font color="Maroon" title="c41">ik</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E15:50_20_1"><i><font color="Green" title="E94">A69</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">4,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c39">k</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E17:50_20_1"><i><font color="Green" title="E96">A71</font></i></a>, <a class="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default">4,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c39">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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="E3:50_20"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c39">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c39">j1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<a NAME="E71:50"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E72:50"/><b>then </b><i><font color="Green" title="E88">A72</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T148" title="XREAL_1:th.148">XREAL_1:148</a></i>;<br/>
<b>set </b><font color="Maroon" title="c40">sp5</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>;<br/>
<a NAME="E73:50"/>
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#E69:50"><i><font color="Green" title="E86">A61</font></i></a>, <a class="txt" href="turing_1.html#E70:50"><i><font color="Green" title="E87">A62</font></i></a>);<br/></i>
<b>then </b><i><font color="Green" title="E89">A73</font></i>: <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c36">p4</font>,<font color="Maroon" title="c37">m2</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default">4,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E72:50"><i><font color="Green" title="E88">A72</font></i></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="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>

;<br/>
<div><i><font color="Green" title="E90">A74</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_22"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon" title="c41">tt</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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 <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a>  ;<br/><i><font color="Green" title="E91">A75</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="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c41">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="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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="turing_1.html#D15" target="_self" title="TURING_1:def.15">Def15</a></i>
<br/>.= 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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">4,<span class="p2">(<span class="default"><font color="Maroon" title="c41">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="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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/>.= 
<a href="turing_1.html#K13" title="TURING_1:func.13">Sum_Tran</a>  <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">4,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span> <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="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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">5,0,0</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#E62:50"><i><font color="Green" title="E81">A54</font></i></a>, <a class="ref" href="turing_1.html#T17" target="_self" title="TURING_1:th.17">Th17</a>, <a class="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
;<br/><i><font color="Green" title="E92">A76</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="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></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="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c10">h1</font>,<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="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
<br/>.= 
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font>
<b>by </b><i><a class="txt" href="turing_1.html#E2:50_22"><i><font color="Green" title="E91">A75</font></i></a>, <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> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font>
<b>by </b><i><a class="txt" href="turing_1.html#E62:50"><i><font color="Green" title="E81">A54</font></i></a>, <a class="ref" href="turing_1.html#T28" target="_self" title="TURING_1:th.28">Th28</a></i>
;<br/><a NAME="E4:50_22"/><i><font color="Green" title="E93">A77</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="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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> 0
 <b>by </b><i><a class="txt" href="turing_1.html#E2:50_22"><i><font color="Green" title="E91">A75</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> <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="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<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="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E3:50_22"><i><font color="Green" title="E92">A76</font></i></a>, <a class="txt" href="turing_1.html#E50"><i><font color="Green" title="E36">Lm6</font></i></a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default">5,<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="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></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="c36">p4</font>,<font color="Maroon" title="c10">h1</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E2:50_22"><i><font color="Green" title="E91">A75</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="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default">5,<span class="p2">(<span class="default"><font color="Maroon" title="c10">h1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E4:50_22"><i><font color="Green" title="E93">A77</font></i></a>, <a class="txt" href="turing_1.html#E45"><i><font color="Green" title="E32">Lm4</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c41">Rs</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>;<br/>
<span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c15">i2</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E11:50"><i><font color="Green" title="E45">A6</font></i></a>, <a class="txt" href="turing_1.html#E29:50"><i><font color="Green" title="E58">A22</font></i></a>, <a class="ref" href="turing_1.html#T12" target="_self" title="TURING_1:th.12">Th12</a></i>
;<br/>
<a NAME="E77:50"/><b>then </b>
<span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c15">i2</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self" title="TURING_1:th.13">Th13</a></i>;<br/>
<b>then </b><i><font color="Green" title="E91">A78</font></i>: <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
 <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c15">i2</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>
<br/>.= 
 <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p4"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c15">i2</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>
<br/>.= 
 <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p0"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c15">i2</font>,<span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self" title="TURING_1:th.13">Th13</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_25"><b>now </b></a><div class="add"><span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p3">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c15">i2</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E78:50"><i><font color="Green" title="E91">A78</font></i></a>, <a class="ref" href="turing_1.html#T13" target="_self" title="TURING_1:th.13">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p3">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p0">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c15">i2</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">n1</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>
;<br/><b>hence </b><a NAME="E2:50_25"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <span class="p2">(<span class="default"><a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p3"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c30">p3</font>,<font color="Maroon" title="c31">m3</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="turing_1.html#E30:50"><i><font color="Green" title="E59">A23</font></i></a>, <a class="txt" href="turing_1.html#E31:50"><i><font color="Green" title="E60">A24</font></i></a>, <a class="txt" href="turing_1.html#E32:50"><i><font color="Green" title="E61">A25</font></i></a>, <a class="txt" href="turing_1.html#E44:50"><i><font color="Green" title="E69">A38</font></i></a>, <a class="txt" href="turing_1.html#E48:50"><i><font color="Green" title="E72">A42</font></i></a>, <a class="txt" href="turing_1.html#E52:50"><i><font color="Green" title="E74">A44</font></i></a>, <a class="txt" href="turing_1.html#E52"><i><font color="Green" title="E38">Lm7</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E80:50"/><b>then </b><i><font color="Green" title="E92">A79</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</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">5,<font color="Maroon" title="c10">h1</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font></span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="turing_1.html#E68:50"><i><font color="Green" title="E85">A60</font></i></a>, <a class="txt" href="turing_1.html#E74:50"><i><font color="Green" title="E89">A73</font></i></a>, <a class="txt" href="turing_1.html#E75:50"><i><font color="Green" title="E90">A74</font></i></a>, <a class="ref" href="turing_1.html#D8" target="_self" title="TURING_1:def.8">Def8</a></i>;<br/>
<b>then </b><i><font color="Green" title="E93">A80</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  = 
5
<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#U5" title="TURING_1:sel.5">AcceptS</a> of <a href="turing_1.html#K14" title="TURING_1:func.14">SumTuring</a> 
<b>by </b><i><a class="ref" href="turing_1.html#D15" target="_self" title="TURING_1:def.15">Def15</a></i>
;<br/>
<b>hence </b><a NAME="E82:50"/>
<font color="Maroon" title="c1">s</font> is <a href="turing_1.html#V2" title="TURING_1:attr.2">Accept-Halt</a>
 <b>by </b><i><a class="ref" href="turing_1.html#D9" target="_self" title="TURING_1:def.9">Def9</a></i>;<br/>

<a NAME="E83:50"/><b>then </b><i><font color="Green" title="E94">A81</font></i>: 
 <a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" title="TURING_1:func.10">Computation</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="turing_1.html#E81:50"><i><font color="Green" title="E93">A80</font></i></a>, <a class="ref" href="turing_1.html#D10" target="_self" title="TURING_1:def.10">Def10</a></i>;<br/>
<b>hence </b><a NAME="E84:50"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">h</font>
 <b>by </b><i><a class="txt" href="turing_1.html#E80:50"><i><font color="Green" title="E92">A79</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>;<br/>

<a NAME="E85:50"/><i><font color="Green" title="E95">A82</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font>
 
<b>by </b><i><a class="txt" href="turing_1.html#E80:50"><i><font color="Green" title="E92">A79</font></i></a>, <a class="txt" href="turing_1.html#E83:50"><i><font color="Green" title="E94">A81</font></i></a>, <a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></i>;<br/>
<a NAME="E86:50"/>
 <a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" title="TURING_1:func.5">Tape-Chg</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c10">h1</font>,<font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c17">nk</font>,<font color="Maroon" title="c8">T</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>,<font color="Maroon" title="c7">F</font> <a href="turing_1.html#R1" title="TURING_1:pred.1">is_1_between</a> <font color="Maroon" title="c10">h1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">h1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2
 
<b>by </b><i><a class="txt" href="turing_1.html#E62:50"><i><font color="Green" title="E81">A54</font></i></a>, <a class="ref" href="turing_1.html#D13" target="_self" title="TURING_1:def.13">Def13</a></i>;<br/>
<b>hence </b><a NAME="E87:50"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" title="TURING_1:func.11">Result</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">h</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="turing_1.html#E85:50"><i><font color="Green" title="E95">A82</font></i></a>, <a class="ref" href="turing_1.html#T19" target="_self" title="TURING_1:th.19">Th19</a></i>;<br/>


</div>
