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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="turing_1.html#NM4">All-State</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="turing_1.html#NM2">Tape</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:62"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="turing_1.html#R2" target="_self">storeData</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> )
 ;<br/>

<a NAME="E2:62"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2 holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) )
 
<b>by </b><i><a class="ref" href="turing_1.html#T20" target="_self">Th20</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="mcart_1.html#K7">`3</a>  as   <a href="turing_1.html#NM2">Tape</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = 0 as   <a href="turing_1.html#NM3">Symbol</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E58"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = 1 as   <a href="turing_1.html#NM3">Symbol</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E58"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<i><font color="Green">E50</font></i>:  <a href="turing_1.html#K8" target="_self">TRAN</a> <font color="Maroon">c<sub>1</sub></font> = 
the <a href="turing_1.html#U3" target="_self">Tran</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</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">c<sub>1</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<a href="turing_1.html#K15" target="_self">Succ_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">c<sub>1</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#D18" target="_self">Def18</a></i>
<br/>.= 
<a href="turing_1.html#K15" target="_self">Succ_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">c<sub>5</sub></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">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:62"><i><font color="Green">E48</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<a href="turing_1.html#K15" target="_self">Succ_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">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:62"><i><font color="Green">E48</font></i></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">1,0,1</span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:62"><i><font color="Green">E48</font></i></a>, <a class="txt" href="turing_1.html#E2:62"><i><font color="Green">E49</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a>, <a class="ref" href="turing_1.html#T34" target="_self">Th34</a></i>
;<br/>
<i><font color="Green">E51</font></i>:  <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></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">c<sub>1</sub></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">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:62"><i><font color="Green">E48</font></i></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> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:62"><i><font color="Green">E48</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E6:62"><i><font color="Green">E50</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E2:62"><i><font color="Green">E49</font></i></a>, <a class="ref" href="turing_1.html#T28" target="_self">Th28</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = 1 as   <a href="turing_1.html#NM1">State</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E57"><i><font color="Green">Lemma44</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>3</sub></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/>
<i><font color="Green">E52</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">c<sub>1</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> 

<br/>.= 
1
<b>by </b><i><a class="txt" href="turing_1.html#E6:62"><i><font color="Green">E50</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span>;<br/>
<i><font color="Green">E53</font></i>:  <a href="turing_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:62"><i><font color="Green">E48</font></i></a>, <a class="txt" href="turing_1.html#E7:62"><i><font color="Green">E51</font></i></a>, <a class="txt" href="turing_1.html#E59"><i><font color="Green">Lemma46</font></i></a></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> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E6:62"><i><font color="Green">E50</font></i></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:62"><i><font color="Green">E48</font></i></a>, <a class="txt" href="turing_1.html#E10:62"><i><font color="Green">E52</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
;<br/>
<a NAME="E12:62"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E13:62"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E14:62"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></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="E15:62"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></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="E16:62"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="txt" href="turing_1.html#E14:62"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></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#K16" target="_self">SuccTuring</a>  ;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/>
<a NAME="E18:62"/><i><font color="Green">E58</font></i>: 
( <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2 holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:62_5"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="turing_1.html#E2:62"><i><font color="Green">E49</font></i></a>, <a class="txt" href="turing_1.html#E12:62"><i><font color="Green">E54</font></i></a>, <a class="ref" href="turing_1.html#T30" target="_self">Th30</a></i>;<br/>

<b>thus </b><a NAME="E2:62_5"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="turing_1.html#E2:62"><i><font color="Green">E49</font></i></a>, <a class="txt" href="turing_1.html#E14:62"><i><font color="Green">E55</font></i></a>, <a class="txt" href="turing_1.html#E15:62"><i><font color="Green">E56</font></i></a>, <a class="ref" href="turing_1.html#T30" target="_self">Th30</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>assume </b><a NAME="E1:62_5_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2 )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_5_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_5_1_1_1"><b>suppose </b></a><a NAME="E1:62_5_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:62_5_1_1_1"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="turing_1.html#T30" target="_self">Th30</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_5_1_1_2"><b>suppose </b></a><a NAME="E1:62_5_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></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> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="turing_1.html#T30" target="_self">Th30</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="turing_1.html#E1:62"><i><font color="Green">E48</font></i></a>, <a class="txt" href="turing_1.html#E1:62_5_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="turing_1.html#T20" target="_self">Th20</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">E59</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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span> = 
the <a href="turing_1.html#U3" target="_self">Tran</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>11</sub></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<a href="turing_1.html#K15" target="_self">Succ_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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>11</sub></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></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="turing_1.html#D18" target="_self">Def18</a></i>
<br/>.= 
<a href="turing_1.html#K15" target="_self">Succ_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></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#K15" target="_self">Succ_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></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#K15" target="_self">Succ_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default">1,1,1</span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:62"><i><font color="Green">E48</font></i></a>, <a class="txt" href="turing_1.html#E12:62"><i><font color="Green">E54</font></i></a>, <a class="txt" href="turing_1.html#E16:62"><i><font color="Green">E57</font></i></a>, <a class="ref" href="turing_1.html#T20" target="_self">Th20</a>, <a class="ref" href="turing_1.html#T34" target="_self">Th34</a></i>
;<br/>
<i><font color="Green">E60</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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></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="turing_1.html#T27" target="_self">Th27</a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E19:62"><i><font color="Green">E59</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/>
<i><font color="Green">E61</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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> = 
<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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K7">`3</a> 

<br/>.= 
1
<b>by </b><i><a class="txt" href="turing_1.html#E19:62"><i><font color="Green">E59</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = 1 as   <a href="turing_1.html#NM1">State</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E57"><i><font color="Green">Lemma44</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>9</sub></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">c<sub>15</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>;<br/>
<i><font color="Green">E62</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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E20:62"><i><font color="Green">E60</font></i></a>, <a class="txt" href="turing_1.html#E59"><i><font color="Green">Lemma46</font></i></a></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E19:62"><i><font color="Green">E59</font></i></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E21:62"><i><font color="Green">E61</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
;<br/>
<a NAME="E25:62"/><i><font color="Green">E63</font></i>: 
the <a href="turing_1.html#U3" target="_self">Tran</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>13</sub></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">c<sub>13</sub></font>,1,1</span><a href="domain_1.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="turing_1.html#D18" target="_self">Def18</a>, <a class="ref" href="turing_1.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E26:62"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>13</sub></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#K16" target="_self">SuccTuring</a> 
 
<b>by </b><i><a class="ref" href="turing_1.html#D18" target="_self">Def18</a></i>;<br/>
<a NAME="E27:62"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">b<sub>1</sub></font> &amp; <font color="Olive">b<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"><font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<b>assume </b><a NAME="E1:62_10"/><i><font color="Green">E65</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></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">c<sub>3</sub></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">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E2:62_10"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></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="E3:62_10"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="turing_1.html#E1:62_10"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E4:62_10"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="turing_1.html#E12:62"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E5:62_10"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="turing_1.html#E18:62"><i><font color="Green">E58</font></i></a>, <a class="txt" href="turing_1.html#E1:62_10"><i><font color="Green">E65</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:62"/><b>then </b><i><font color="Green">E65</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">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="turing_1.html#E25:62"><i><font color="Green">E63</font></i></a>, <a class="txt" href="turing_1.html#E26:62"><i><font color="Green">E64</font></i></a>, <a class="txt" href="turing_1.html#E49"><i><font color="Green">Lemma38</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></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">c<sub>17</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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#K16" target="_self">SuccTuring</a>  ;<br/>
<i><font color="Green">E66</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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span> = 
the <a href="turing_1.html#U3" target="_self">Tran</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>18</sub></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<a href="turing_1.html#K15" target="_self">Succ_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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>18</sub></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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="turing_1.html#D18" target="_self">Def18</a></i>
<br/>.= 
<a href="turing_1.html#K15" target="_self">Succ_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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#K15" target="_self">Succ_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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="txt" href="turing_1.html#E18:62"><i><font color="Green">E58</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a>, <a class="ref" href="turing_1.html#T34" target="_self">Th34</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>19</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/>
<a NAME="E32:62"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>3</sub></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">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E33:62"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></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="E34:62"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="turing_1.html#E32:62"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E35:62"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="turing_1.html#E12:62"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E36:62"/><i><font color="Green">E70</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2 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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_12"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:62_12"/>
<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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="turing_1.html#E12:62"><i><font color="Green">E54</font></i></a>, <a class="txt" href="turing_1.html#E18:62"><i><font color="Green">E58</font></i></a>, <a class="txt" href="turing_1.html#E34:62"><i><font color="Green">E68</font></i></a>, <a class="ref" href="turing_1.html#T30" target="_self">Th30</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>20</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>assume </b><a NAME="E1:62_12_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2 )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_12_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2 or <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_12_1_1_1"><b>suppose </b></a><a NAME="E1:62_12_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 ;<br/><div class="add"><a NAME="E2:62_12_1_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="txt" href="turing_1.html#E1:62_12_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</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> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> = 
<span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E1:62_12_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="turing_1.html#T30" target="_self">Th30</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="turing_1.html#E18:62"><i><font color="Green">E58</font></i></a>, <a class="txt" href="turing_1.html#E1:62_12_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="turing_1.html#E2:62_12_1_1_1"><i><font color="Green">E73</font></i></a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_12_1_1_2"><b>suppose </b></a><a NAME="E1:62_12_1_1_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:62_12_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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="turing_1.html#T30" target="_self">Th30</a></i>;<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">E71</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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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="turing_1.html#T27" target="_self">Th27</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E31:62"><i><font color="Green">E66</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/>
<i><font color="Green">E72</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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> = 
<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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K7">`3</a> 

<br/>.= 
1
<b>by </b><i><a class="txt" href="turing_1.html#E31:62"><i><font color="Green">E66</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></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">c<sub>22</sub></font> = 2 as   <a href="turing_1.html#NM1">State</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E57"><i><font color="Green">Lemma44</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = 3 as   <a href="turing_1.html#NM1">State</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E57"><i><font color="Green">Lemma44</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>24</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>;<br/>
<i><font color="Green">E73</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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E37:62"><i><font color="Green">E71</font></i></a>, <a class="txt" href="turing_1.html#E59"><i><font color="Green">Lemma46</font></i></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E31:62"><i><font color="Green">E66</font></i></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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E38:62"><i><font color="Green">E72</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></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#K16" target="_self">SuccTuring</a>  ;<br/>
<a NAME="E44:62"/><i><font color="Green">E74</font></i>: 
the <a href="turing_1.html#U1" target="_self">Symbols</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="turing_1.html#D18" target="_self">Def18</a></i>;<br/>
<div><i><font color="Green">E75</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_16"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_16_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_16_1"/>
( <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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 1 or <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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i><a class="txt" href="turing_1.html#E44:62"><i><font color="Green">E74</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_16_1_1"><b>suppose </b></a><a NAME="E1:62_16_1_1"/>
<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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:62_16_1_1"/>
<a href="turing_1.html#K15" target="_self">Succ_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">2,<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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,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="ref" href="turing_1.html#T34" target="_self">Th34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_16_1_2"><b>suppose </b></a><a NAME="E1:62_16_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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:62_16_1_2"/>
<a href="turing_1.html#K15" target="_self">Succ_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">2,<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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,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="ref" href="turing_1.html#T34" target="_self">Th34</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<i><font color="Green">E76</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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span> = 
the <a href="turing_1.html#U3" target="_self">Tran</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>25</sub></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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<a href="turing_1.html#K15" target="_self">Succ_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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>25</sub></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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></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="turing_1.html#D18" target="_self">Def18</a></i>
<br/>.= 
<a href="turing_1.html#K15" target="_self">Succ_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon">c<sub>25</sub></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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></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#K15" target="_self">Succ_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></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"><font color="Maroon">c<sub>23</sub></font>,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#E45:62"><i><font color="Green">E75</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>26</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<i><font color="Green">E77</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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></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="turing_1.html#T27" target="_self">Th27</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E46:62"><i><font color="Green">E76</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/>
<i><font color="Green">E78</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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> = 
<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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K7">`3</a> 

<br/>.= 
 <a href="real_1.html#K1">-</a> 1
<b>by </b><i><a class="txt" href="turing_1.html#E46:62"><i><font color="Green">E76</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/>
<a NAME="E49:62"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="xcmplx_0.html#K2">+</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"><font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font>
 
;<br/>
<b>set </b><font color="Maroon">c<sub>27</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>16</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>;<br/>
<i><font color="Green">E80</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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E47:62"><i><font color="Green">E77</font></i></a>, <a class="txt" href="turing_1.html#E59"><i><font color="Green">Lemma46</font></i></a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E46:62"><i><font color="Green">E76</font></i></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>16</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E48:62"><i><font color="Green">E78</font></i></a>, <a class="txt" href="turing_1.html#E49:62"><i><font color="Green">E79</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
;<br/>
<a NAME="E51:62"/><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</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"><font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></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="E52:62"/><i><font color="Green">E82</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_21"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:62_21"/>
<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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="turing_1.html#E35:62"><i><font color="Green">E69</font></i></a>, <a class="txt" href="turing_1.html#E36:62"><i><font color="Green">E70</font></i></a>, <a class="txt" href="turing_1.html#E51:62"><i><font color="Green">E81</font></i></a>, <a class="ref" href="turing_1.html#T30" target="_self">Th30</a></i>;<br/>

<b>thus </b><a NAME="E2:62_21"/>
<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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="turing_1.html#T30" target="_self">Th30</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>28</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>assume </b><a NAME="E1:62_21_1"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 )
 ;<br/><a NAME="E2:62_21_1"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E3:62_21_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E1:62_21_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="turing_1.html#T30" target="_self">Th30</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="turing_1.html#E36:62"><i><font color="Green">E70</font></i></a>, <a class="txt" href="turing_1.html#E1:62_21_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="turing_1.html#E3:62_21_1"><i><font color="Green">E84</font></i></a></i>
;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> 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">c<sub>23</sub></font>,<font color="Maroon">c<sub>16</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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="mcart_1.html#K3">[</a><span class="default">3,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xcmplx_0.html#K6">-</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span> );<br/>
<a NAME="E53:62"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E54:62"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_22"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>28</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:62_22"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>28</sub></font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_22_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_22_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><a NAME="E2:62_22_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></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">c<sub>29</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>28</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>28</sub></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">c<sub>31</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>;<br/><b>reconsider </b><font color="Maroon">c<sub>32</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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#K16" target="_self">SuccTuring</a>  ;<br/><a NAME="E5:62_22_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>16</sub></font> <a href="xcmplx_0.html#K2">+</a> 0
 <b>by </b><i><a class="txt" href="turing_1.html#E1:62_22_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="turing_1.html#E2:62_22_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E6:62_22_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>16</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T23">XREAL_1:23</a></i>;<br/><a NAME="E7:62_22_1"/>
<font color="Maroon">c<sub>3</sub></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="E8:62_22_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="turing_1.html#E6:62_22_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>28</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E10:62_22_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="ref" href="int_1.html#T19">INT_1:19</a></i>;<br/><a NAME="E11:62_22_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_22_1_1"><b>now </b></a><div class="add"><a NAME="E1:62_22_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</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>hence </b><a NAME="E2:62_22_1_1"/>
<font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="txt" href="turing_1.html#E11:62_22_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div><a NAME="E13:62_22_1"/><b>then </b><i><font color="Green">E90</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="turing_1.html#E52:62"><i><font color="Green">E82</font></i></a>, <a class="txt" href="turing_1.html#E6:62_22_1"><i><font color="Green">E88</font></i></a></i>;<br/><div><i><font color="Green">E91</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_22_1_2"><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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span> = 
the <a href="turing_1.html#U3" target="_self">Tran</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>32</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<a href="turing_1.html#K15" target="_self">Succ_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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>32</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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="turing_1.html#D18" target="_self">Def18</a></i>
<br/>.= 
<a href="turing_1.html#K15" target="_self">Succ_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>32</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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#K15" target="_self">Succ_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">3,1,<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#E13:62_22_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a>, <a class="ref" href="turing_1.html#T34" target="_self">Th34</a></i>
;<br/></div><b>end;</b></div><div><i><font color="Green">E92</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_22_1_3"><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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>28</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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="turing_1.html#T27" target="_self">Th27</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E14:62_22_1"><i><font color="Green">E91</font></i></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> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E13:62_22_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="turing_1.html#T28" target="_self">Th28</a></i>
;<br/></div><b>end;</b></div><i><font color="Green">E93</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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> = 
<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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K7">`3</a> 

<br/>.= 
 <a href="real_1.html#K1">-</a> 1
<b>by </b><i><a class="txt" href="turing_1.html#E14:62_22_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/><div><b>hereby </b>
<div class="add"><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">c<sub>23</sub></font>,<font color="Maroon">c<sub>16</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>28</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:62_22"><i><font color="Green">E85</font></i></a>, <a class="txt" href="turing_1.html#E1:62_22_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="turing_1.html#E2:62_22_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="turing_1.html#D8" target="_self">Def8</a>, <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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E15:62_22_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="turing_1.html#E59"><i><font color="Green">Lemma46</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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E14:62_22_1"><i><font color="Green">E91</font></i></a>, <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">3,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E16:62_22_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default">3,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>

;<br/></div>
<b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:62_22"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>3</sub></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">c<sub>29</sub></font> = <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>30</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>;<br/>
<a NAME="E56:62"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 
;<br/>
<a NAME="E57:62"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="turing_1.html#E53:62"><i><font color="Green">E83</font></i></a>, <a class="txt" href="turing_1.html#E54:62"><i><font color="Green">E84</font></i></a>);<br/></i>
<b>then </b><i><font color="Green">E86</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">c<sub>23</sub></font>,<font color="Maroon">c<sub>16</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default">3,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E56:62"><i><font color="Green">E85</font></i></a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>

;<br/>
<div><i><font color="Green">E87</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_24"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon">c<sub>31</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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#K16" target="_self">SuccTuring</a>  ;<br/><i><font color="Green">E88</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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span> = 
the <a href="turing_1.html#U3" target="_self">Tran</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>31</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<a href="turing_1.html#K15" target="_self">Succ_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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>31</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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="turing_1.html#D18" target="_self">Def18</a></i>
<br/>.= 
<a href="turing_1.html#K15" target="_self">Succ_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<span class="p2">(<span class="default"><font color="Maroon">c<sub>31</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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#K15" target="_self">Succ_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">4,0,0</span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E52:62"><i><font color="Green">E82</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a>, <a class="ref" href="turing_1.html#T34" target="_self">Th34</a></i>
;<br/><i><font color="Green">E89</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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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="turing_1.html#T27" target="_self">Th27</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E2:62_24"><i><font color="Green">E88</font></i></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> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E52:62"><i><font color="Green">E82</font></i></a>, <a class="ref" href="turing_1.html#T28" target="_self">Th28</a></i>
;<br/><i><font color="Green">E90</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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> = 
<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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K7">`3</a> 

<br/>.= 
0
<b>by </b><i><a class="txt" href="turing_1.html#E2:62_24"><i><font color="Green">E88</font></i></a>, <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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E3:62_24"><i><font color="Green">E89</font></i></a>, <a class="txt" href="turing_1.html#E59"><i><font color="Green">Lemma46</font></i></a></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>28</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E2:62_24"><i><font color="Green">E88</font></i></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">4,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E4:62_24"><i><font color="Green">E90</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>31</sub></font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</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">c<sub>4</sub></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>;<br/>
<span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</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">c<sub>4</sub></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></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"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</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">c<sub>4</sub></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self">Th13</a></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"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> 1</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">c<sub>4</sub></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self">Def8</a></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"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font></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">c<sub>4</sub></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E11:62"><i><font color="Green">E53</font></i></a>, <a class="ref" href="turing_1.html#T12" target="_self">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>4</sub></font> <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="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E24:62"><i><font color="Green">E62</font></i></a>, <a class="ref" href="turing_1.html#T13" target="_self">Th13</a></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"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<span class="p0">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</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">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self">Def8</a></i>
;<br/>
<a NAME="E61:62"/><b>then </b>
<span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</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">c<sub>4</sub></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></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">c<sub>23</sub></font>,<font color="Maroon">c<sub>16</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></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">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="turing_1.html#E28:62"><i><font color="Green">E65</font></i></a>, <a class="txt" href="turing_1.html#E42:62"><i><font color="Green">E73</font></i></a>, <a class="txt" href="turing_1.html#E50:62"><i><font color="Green">E80</font></i></a>, <a class="ref" href="turing_1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E62:62"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</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">c<sub>4</sub></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default">4,<font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="turing_1.html#E58:62"><i><font color="Green">E86</font></i></a>, <a class="txt" href="turing_1.html#E59:62"><i><font color="Green">E87</font></i></a>, <a class="ref" href="turing_1.html#D8" target="_self">Def8</a></i>;<br/>
<b>then </b><i><font color="Green">E89</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">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</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">c<sub>4</sub></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a>  = 
4
<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#K16" target="_self">SuccTuring</a> 
<b>by </b><i><a class="ref" href="turing_1.html#D18" target="_self">Def18</a></i>
;<br/>
<b>hence </b><a NAME="E64:62"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="turing_1.html#V2" target="_self">Accept-Halt</a>
 <b>by </b><i><a class="ref" href="turing_1.html#D9" target="_self">Def9</a></i>;<br/>

<a NAME="E65:62"/><b>then </b><i><font color="Green">E90</font></i>: 
 <a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</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">c<sub>4</sub></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="turing_1.html#E63:62"><i><font color="Green">E89</font></i></a>, <a class="ref" href="turing_1.html#D10" target="_self">Def10</a></i>;<br/>
<b>hence </b><a NAME="E66:62"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="turing_1.html#E62:62"><i><font color="Green">E88</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/>

<a NAME="E67:62"/><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="turing_1.html#E62:62"><i><font color="Green">E88</font></i></a>, <a class="txt" href="turing_1.html#E65:62"><i><font color="Green">E90</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/>
<a NAME="E68:62"/>
 <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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="turing_1.html#R1" target="_self">is_1_between</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="txt" href="turing_1.html#E52:62"><i><font color="Green">E82</font></i></a>, <a class="ref" href="turing_1.html#D13" target="_self">Def13</a></i>;<br/>
<b>hence </b><a NAME="E69:62"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">c<sub>1</sub></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"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="turing_1.html#E67:62"><i><font color="Green">E91</font></i></a>, <a class="ref" href="turing_1.html#T19" target="_self">Th19</a></i>;<br/>


</div>
