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<b>let </b><font color="Maroon">s</font> be   <a href="turing_1.html#NM4">All-State</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a> ;<br/><b>let </b><font color="Maroon">t</font> be   <a href="turing_1.html#NM2">Tape</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a> ;<br/><b>let </b><font color="Maroon">h</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:52"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default">0,<font color="Maroon">h</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span> &amp; <font color="Maroon">t</font> <a href="turing_1.html#R2" target="_self">storeData</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">h</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font></span><a href="finseq_4.html#K3">*&gt;</a></span> )
 ;<br/>

<b>reconsider </b><font color="Maroon">s3</font> = <font color="Maroon">s</font> <a href="mcart_1.html#K7">`3</a>  as   <a href="turing_1.html#NM2">Tape</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  ;<br/>
<b>reconsider </b><font color="Maroon">F</font> = 0 as   <a href="turing_1.html#NM3">Symbol</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E7:52"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">T</font> = 1 as   <a href="turing_1.html#NM3">Symbol</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E7:52"><i><font color="Green">E34</font></i></a></i>;<br/>
<i><font color="Green">E31</font></i>:  <a href="turing_1.html#K8" target="_self">TRAN</a> <font color="Maroon">s</font> = 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s3</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <font color="Maroon">s</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:52"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">s3</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <font color="Maroon">s</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <font color="Maroon">s</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default">0,0,1</span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i>, , </i>
;<br/>
<a NAME="E6:52"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E34</font></i>:  <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="mcart_1.html#K7">`3</a> </span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <font color="Maroon">s</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> = 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <font color="Maroon">s</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <font color="Maroon">s</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i>, <a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h</font>,<font color="Maroon">F</font>
<b>by </b><i><a class="ref" href="turing_1.html#T1" target="_self">Th1</a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<font color="Maroon">t</font>
<b>by </b><i><a class="ref" href="turing_1.html#T2" target="_self">Th2</a>, </i>
;<br/>
<b>reconsider </b><font color="Maroon">p0</font> = 0 as   <a href="turing_1.html#NM1">State</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E5:52"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<a NAME="E10:52"/><i><font color="Green">E35</font></i>: 
 <a href="turing_1.html#K6" target="_self">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="turing_1.html#T1" target="_self">Th1</a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/>
<b>set </b><font color="Maroon">s1</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span>;<br/>
<i><font color="Green">E75</font></i>:  <a href="turing_1.html#K9" target="_self">Following</a> <font color="Maroon">s</font> = 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <font color="Maroon">s</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">t</font></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">t</font></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#T1" target="_self">Th1</a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i>, , <a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</font></i></a></i>
;<br/>
<b>reconsider </b><font color="Maroon">s3</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K7">`3</a>  as   <a href="turing_1.html#NM2">Tape</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  ;<br/>
<a NAME="E13:52"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a>  st <font color="Maroon">h</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 holds <br/><font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( for <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a>  st <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4 holds <br/><font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1 ) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:52"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">h</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h1</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E15:52"/>
<font color="Maroon">h</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E16:52"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E17:52"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">h1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E18:52"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">h1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E19:52"/><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
;<br/>
<b>set </b><font color="Maroon">t1</font> =  <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font>;<br/>
<a NAME="E20:52"/>
<font color="Maroon">h</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E21:52"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 4 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E22:52"/><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E23:52"/><i><font color="Green">E126</font></i>: 
( <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a>  st <font color="Maroon">h1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( for <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a>  st <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4 holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_4"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="turing_1.html#T3" target="_self">Th3</a>, </i>;<br/>

<b>thus </b><a NAME="E2:52_4"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , , </i>;<br/>

<a NAME="E3:52_4"/>
<font color="Maroon">h1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 4
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E4:52_4"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="turing_1.html#T4" target="_self">Th4</a>, </i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>assume </b><a NAME="E1:52_4_1"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">h1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> )
 ;<br/><a NAME="E2:52_4_1"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">h</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="turing_1.html#T3" target="_self">Th3</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="turing_1.html#T7" target="_self">Th7</a>, </i>
<br/>.= 
1
<b>by </b><i>, , <a class="ref" href="turing_1.html#T7" target="_self">Th7</a>, , </i>
;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>assume </b><a NAME="E1:52_4_2"/><i><font color="Green">E130</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4 )
 ;<br/><b>hence </b><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, </i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="turing_1.html#T7" target="_self">Th7</a>, </i>
;<br/><br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<i><font color="Green">E131</font></i>:  <a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span> = 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s3</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:52"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<span class="p2">(<span class="default"><font color="Maroon">s3</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">h1</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default">1,0,1</span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="turing_1.html#T3" target="_self">Th3</a>, , , </i>
;<br/>
<i><font color="Green">E160</font></i>:  <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K7">`3</a> </span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> = 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font>
<b>by </b><i><a class="ref" href="turing_1.html#T7" target="_self">Th7</a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/>
<a NAME="E26:52"/><i><font color="Green">E161</font></i>: 
 <a href="turing_1.html#K6" target="_self">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="turing_1.html#T7" target="_self">Th7</a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p1</font> = 1 as   <a href="turing_1.html#NM1">State</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E5:52"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">i2</font> = <font color="Maroon">h1</font> <a href="xcmplx_0.html#K2">+</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">s2</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">i2</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span>;<br/>
<i><font color="Green">E162</font></i>:  <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span> = 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default">1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p0</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#T7" target="_self">Th7</a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">i2</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i>, <a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</font></i></a></i>
;<br/>
<a NAME="E30:52"/><i><font color="Green">E163</font></i>: 
the <a href="turing_1.html#U3" target="_self">Tran</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p1</font>,1</span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,1,1</span><a href="domain_1.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="turing_1.html#E1:52"><i><font color="Green">E30</font></i></a>, </i>;<br/>
<a NAME="E31:52"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a> 
 
<b>by </b><i><a class="txt" href="turing_1.html#E1:52"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E32:52"/><i><font color="Green">E165</font></i>: 
for <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a>  st <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<b>assume </b><a NAME="E1:52_8"/><i><font color="Green">E166</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> )
 ;<br/>

<a NAME="E2:52_8"/>
<font color="Maroon">h1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h1</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E3:52_8"/><b>then </b>
<font color="Maroon">h1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E4:52_8"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="turing_1.html#T6" target="_self">Th6</a>, </i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">nk</font> = <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">sn</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">nk</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">sn3</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">nk</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K7">`3</a>  as   <a href="turing_1.html#NM2">Tape</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  ;<br/>
<div><i><font color="Green">E170</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_9"><b>now </b></a><div class="add"><b>thus </b> <a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">nk</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> = 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">nk</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">sn3</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">nk</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:52"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<span class="p2">(<span class="default"><font color="Maroon">sn3</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">nk</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">nk</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default">2,1,1</span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#T6" target="_self">Th6</a>, , <a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">t2</font> =  <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font>;<br/>
<a NAME="E36:52"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">h1</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E37:52"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">h1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h1</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E38:52"/><b>then </b><i><font color="Green">E174</font></i>: 
<font color="Maroon">h1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n1</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E39:52"/><i><font color="Green">E175</font></i>: 
( <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a>  st <font color="Maroon">h1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4 holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_10"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_10"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="turing_1.html#T5" target="_self">Th5</a>, , , </i>;<br/>

<a NAME="E2:52_10"/>
<font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E3:52_10"/><b>then </b><i><font color="Green">E176</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E4:52_10"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E5:52_10"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="turing_1.html#T6" target="_self">Th6</a>, <a class="ref" href="turing_1.html#D7" target="_self">Def7</a>, </i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>assume </b><a NAME="E1:52_10_1"/><i><font color="Green">E177</font></i>: 
( <font color="Maroon">h1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4 )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_10_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_10_1_1_1"><b>suppose </b></a><a NAME="E1:52_10_1_1_1"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i/>
<br/>.= 
1
<b>by </b><i><a class="ref" href="turing_1.html#T6" target="_self">Th6</a>, <a class="ref" href="turing_1.html#D8" target="_self">Def8</a>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_10_1_1_2"><b>suppose </b></a><a NAME="E1:52_10_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_10_1_1_2"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_10_1_1_3"><b>suppose </b></a><a NAME="E1:52_10_1_1_3"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i/>
<br/>.= 
1
<b>by </b><i><a class="ref" href="turing_1.html#T6" target="_self">Th6</a>, <a class="ref" href="turing_1.html#D8" target="_self">Def8</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">E180</font></i>:  <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">nk</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K7">`3</a> </span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">nk</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">nk</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> = 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">nk</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">nk</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">nk</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/>
<a NAME="E41:52"/><i><font color="Green">E181</font></i>: 
 <a href="turing_1.html#K6" target="_self">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">nk</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/>
<b>set </b><font color="Maroon">i3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>reconsider </b><font color="Maroon">n3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p2</font> = 2 as   <a href="turing_1.html#NM1">State</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E5:52"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">sm</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n3</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span>;<br/>
<i><font color="Green">E184</font></i>:  <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">nk</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> = 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">nk</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">nk</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">nk</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#D7" target="_self">Def7</a>, </i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default">2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">nk</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">nk</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n3</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self">Def8</a>, <a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</font></i></a></i>
;<br/>
<a NAME="E45:52"/><i><font color="Green">E185</font></i>: 
the <a href="turing_1.html#U3" target="_self">Tran</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p2</font>,1</span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,1,1</span><a href="domain_1.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="turing_1.html#E1:52"><i><font color="Green">E30</font></i></a>, </i>;<br/>
<a NAME="E46:52"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a> 
 
<b>by </b><i><a class="txt" href="turing_1.html#E1:52"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E47:52"/>
for <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&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">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<b>assume </b><a NAME="E1:52_13"/><i><font color="Green">E187</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&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">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/>

<a NAME="E2:52_13"/>
<font color="Maroon">nk</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E3:52_13"/><b>then </b>
<font color="Maroon">h1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E4:52_13"/><b>then </b>
<font color="Maroon">h1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="turing_1.html#T12" target="_self">Th12</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E5:52_13"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="turing_1.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E48:52"/><b>then </b><i><font color="Green">E188</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n3</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span>
 
<b>by </b><i>, , </i>;<br/>
<b>set </b><font color="Maroon">i4</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4;<br/>
<b>reconsider </b><font color="Maroon">n4</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">sp2</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">sn3</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K7">`3</a>  as   <a href="turing_1.html#NM2">Tape</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  ;<br/>
<i><font color="Green">E191</font></i>:  <a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> = 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">sn3</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:52"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<span class="p2">(<span class="default"><font color="Maroon">sn3</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default">3,0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i>, , <a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</font></i></a></i>
;<br/>
<b>set </b><font color="Maroon">j3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1;<br/>
<div><i><font color="Green">E193</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_15"><b>now </b></a><div class="add"><i><font color="Green">E194</font></i>:  <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K7">`3</a> </span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> = 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span>,<font color="Maroon">F</font>
<b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self">Th13</a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font>
<b>by </b><i>, </i>
;<br/><a NAME="E2:52_15"/><i><font color="Green">E195</font></i>: 
 <a href="turing_1.html#K6" target="_self">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self">Th13</a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/><a NAME="E3:52_15"/><i><font color="Green">E196</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><b>thus </b> <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> = 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E11:52"><i><font color="Green">E75</font></i></a>, </i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default">3,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self">Th13</a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#T15" target="_self">Th15</a>, <a class="ref" href="turing_1.html#D9" target="_self">Def9</a>, <a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">p3</font> = 3 as   <a href="turing_1.html#NM1">State</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E5:52"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">m3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">sp3</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">sn3</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K7">`3</a>  as   <a href="turing_1.html#NM2">Tape</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  ;<br/>
<a NAME="E56:52"/><i><font color="Green">E199</font></i>: 
<font color="Maroon">h1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 3
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E57:52"/><i><font color="Green">E200</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="turing_1.html#T4" target="_self">Th4</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E58:52"/><b>then </b><i><font color="Green">E201</font></i>: 
<font color="Maroon">h1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="turing_1.html#E11:52"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E59:52"/><i><font color="Green">E202</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4
 
<b>by </b><i><a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/>
<a NAME="E60:52"/><b>then </b><i><font color="Green">E203</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="turing_1.html#D9" target="_self">Def9</a></i>;<br/>
<div><i><font color="Green">E204</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_16"><b>now </b></a><div class="add"><b>thus </b> <a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> = 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">sn3</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:52"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<span class="p2">(<span class="default"><font color="Maroon">sn3</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default">4,0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i>, , <a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">t3</font> =  <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font>;<br/>
<a NAME="E62:52"/><i><font color="Green">E206</font></i>: 
( <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a>  st <font color="Maroon">h1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1 holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_17"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_17"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="txt" href="turing_1.html#E11:52"><i><font color="Green">E75</font></i></a>, <a class="ref" href="turing_1.html#T15" target="_self">Th15</a>, </i>;<br/>

<b>thus </b><a NAME="E2:52_17"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>assume </b><a NAME="E1:52_17_1"/><i><font color="Green">E207</font></i>: 
( <font color="Maroon">h1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1 )
 ;<br/><a NAME="E2:52_17_1"/><b>then </b><i><font color="Green">E208</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4
 <b>by </b><i><a class="ref" href="turing_1.html#D10" target="_self">Def10</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, </i>
<br/>.= 
1
<b>by </b><i>, , <a class="ref" href="turing_1.html#T16" target="_self">Th16</a></i>
;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<i><font color="Green">E209</font></i>:  <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K7">`3</a> </span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> = 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/>
<a NAME="E64:52"/><i><font color="Green">E210</font></i>: 
 <a href="turing_1.html#K6" target="_self">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/>
<b>set </b><font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1;<br/>
<a NAME="E65:52"/><i><font color="Green">E212</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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="hidden.html#R1">=</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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
;<br/>
<b>reconsider </b><font color="Maroon">p4</font> = 4 as   <a href="turing_1.html#NM1">State</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E5:52"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">sp4</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">m2</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span>;<br/>
<div><i><font color="Green">E215</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_19"><b>now </b></a><div class="add"><b>thus </b> <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> = 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default">4,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">m2</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#T16" target="_self">Th16</a>, , <a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</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">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 implies <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">m2</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> );<br/>
<a NAME="E69:52"/><i><font color="Green">E216</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i/>;<br/>
<a NAME="E70:52"/><i><font color="Green">E217</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_20"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:52_20"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_20_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_20_1"/><i><font color="Green">E219</font></i>: 
<font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E2:52_20_1"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><b>set </b><font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font>;<br/><b>reconsider </b><font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><b>set </b><font color="Maroon">sk</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">ik</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span>;<br/><b>reconsider </b><font color="Maroon">tt</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">ik</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K7">`3</a>  as   <a href="turing_1.html#NM2">Tape</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  ;<br/><a NAME="E5:52_20_1"/>
<font color="Maroon">h1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 0
 <b>by </b><i/>;<br/><a NAME="E6:52_20_1"/><b>then </b><i><font color="Green">E222</font></i>: 
<font color="Maroon">h1</font> <a href="xcmplx_0.html#K6">-</a> 0 <a href="xxreal_0.html#NR3" target="_self">&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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T23">XREAL_1:23</a></i>;<br/><a NAME="E7:52_20_1"/>
<font color="Maroon">h1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E8:52_20_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E10:52_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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="int_1.html#T19">INT_1:19</a></i>;<br/><a NAME="E11:52_20_1"/><b>then </b><i><font color="Green">E224</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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E12:52_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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/><a NAME="E13:52_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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E14:52_20_1"/><b>then </b><i><font color="Green">E225</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, </i>;<br/><div><i><font color="Green">E226</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_20_1_1"><b>now </b></a><div class="add"><b>thus </b> <a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">ik</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> = 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">ik</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">tt</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">ik</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:52"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<span class="p2">(<span class="default"><font color="Maroon">tt</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">ik</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">ik</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default">4,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i>, , <a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</font></i></a></i>
;<br/></div><b>end;</b></div><div><i><font color="Green">E227</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_20_1_2"><b>now </b></a><div class="add"><b>thus </b> <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">ik</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K7">`3</a> </span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">ik</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">ik</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> = 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">ik</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">ik</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">ik</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">T</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div><a NAME="E17:52_20_1"/><i><font color="Green">E228</font></i>: 
 <a href="turing_1.html#K6" target="_self">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">ik</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">m2</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">ik</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i>, , , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">ik</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">ik</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">ik</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default">4,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">ik</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">ik</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i>, <a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span>

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:52_20"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">j1</font> = <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<a NAME="E71:52"/>
<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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E72:52"/><b>then </b><i><font color="Green">E230</font></i>: 
<font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/>
<b>set </b><font color="Maroon">sp5</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span>;<br/>
<a NAME="E73:52"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>then </b><i><font color="Green">E232</font></i>: <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">m2</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1"><a href="domain_1.html#K4">[</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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span>

;<br/>
<div><i><font color="Green">E233</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_22"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon">tt</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K7">`3</a>  as   <a href="turing_1.html#NM2">Tape</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  ;<br/><i><font color="Green">E234</font></i>:  <a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> = 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">tt</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:52"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">4,<span class="p2">(<span class="default"><font color="Maroon">tt</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">4,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default">5,0,0</span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i>, , <a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</font></i></a></i>
;<br/><i><font color="Green">E235</font></i>:  <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K7">`3</a> </span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> = 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">h1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">h1</font>,<font color="Maroon">F</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font>
<b>by </b><i>, </i>
;<br/><a NAME="E4:52_22"/><i><font color="Green">E236</font></i>: 
 <a href="turing_1.html#K6" target="_self">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/><b>thus </b> <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span> = 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default">5,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p4</font>,<font color="Maroon">h1</font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default">5,<span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="xcmplx_0.html#K2">+</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i>, <a class="txt" href="turing_1.html#E6:52"><i><font color="Green">E32</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">Rs</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</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">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/>
<span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">i2</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i>, , </i>
;<br/>
<a NAME="E77:52"/><b>then </b>
<span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</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">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">i2</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</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">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i/>;<br/>
<b>then </b><i><font color="Green">E238</font></i>: <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</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">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
 <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">i2</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1">(<span class="default"><a href="turing_1.html#K9" target="_self">Following</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">i2</font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1">(<span class="default"><a href="turing_1.html#K9" target="_self">Following</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_25"><b>now </b></a><div class="add"><span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</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">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2">(<span class="default"><a href="turing_1.html#K9" target="_self">Following</a> <span class="p3">(<span class="default"><a href="turing_1.html#K9" target="_self">Following</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">i2</font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2">(<span class="default"><a href="turing_1.html#K9" target="_self">Following</a> <span class="p3">(<span class="default"><a href="turing_1.html#K9" target="_self">Following</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0">(<span class="default"><a href="turing_1.html#K9" target="_self">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">i2</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n1</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E2:52_25"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</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">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2">(<span class="default"><a href="turing_1.html#K9" target="_self">Following</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p3</font>,<font color="Maroon">m3</font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="turing_1.html#T8" target="_self">Th8</a>, , , <a class="ref" href="turing_1.html#T12" target="_self">Th12</a>, <a class="ref" href="turing_1.html#T14" target="_self">Th14</a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E80:52"/><b>then </b><i><font color="Green">E239</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</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">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default">5,<font color="Maroon">h1</font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K4">]</a></span>
 
<b>by </b><i>, , , </i>;<br/>
<b>then </b><i><font color="Green">E240</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</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">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a>  = 
5
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a> 
<b>by </b><i><a class="txt" href="turing_1.html#E1:52"><i><font color="Green">E30</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E82:52"/>
<font color="Maroon">s</font> is <a href="turing_1.html#V2" target="_self">Accept-Halt</a>
 <b>by </b><i/>;<br/>

<a NAME="E83:52"/><b>then </b><i><font color="Green">E241</font></i>: 
 <a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</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">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E84:52"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">s</font></span>)</span> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">h</font>
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/>

<a NAME="E85:52"/><i><font color="Green">E242</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">s</font></span>)</span> <a href="mcart_1.html#K7">`3</a>  <a href="hidden.html#R1">=</a>  <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font>
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/>
<a NAME="E86:52"/>
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h1</font>,<font color="Maroon">F</font></span>)</span>,<font color="Maroon">nk</font>,<font color="Maroon">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">h</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">F</font> <a href="turing_1.html#R1" target="_self">is_1_between</a> <font color="Maroon">h1</font>,<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E87:52"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">s</font></span>)</span> <a href="mcart_1.html#K7">`3</a>  <a href="turing_1.html#R2" target="_self">storeData</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">h</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i>, </i>;<br/>


</div>
