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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#K14" target="_self">SumTuring</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#K14" target="_self">SumTuring</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:52"/><i><font color="Green">E40</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#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="finseq_4.html#K3">*&gt;</a></span> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</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#K14" target="_self">SumTuring</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = 0 as   <a href="turing_1.html#NM3">Symbol</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E45"><i><font color="Green">Lemma34</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = 1 as   <a href="turing_1.html#NM3">Symbol</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E45"><i><font color="Green">Lemma34</font></i></a></i>;<br/>
<i><font color="Green">E41</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#K14" target="_self">SumTuring</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>6</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#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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>6</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#D15" target="_self">Def15</a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</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#E1:52"><i><font color="Green">E40</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">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#E1:52"><i><font color="Green">E40</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:52"><i><font color="Green">E40</font></i></a>, <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">0,0,1</span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:52"><i><font color="Green">E40</font></i></a>, <a class="ref" href="turing_1.html#T17" target="_self">Th17</a>, <a class="ref" href="turing_1.html#T22" target="_self">Th22</a></i>
;<br/>
<a NAME="E6:52"/><i><font color="Green">E42</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
 
<b>by </b><i><a class="txt" href="turing_1.html#E1:52"><i><font color="Green">E40</font></i></a>, <a class="ref" href="turing_1.html#T22" target="_self">Th22</a></i>;<br/>
<i><font color="Green">E43</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#E1:52"><i><font color="Green">E40</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#E1:52"><i><font color="Green">E40</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>7</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E5:52"><i><font color="Green">E41</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#E6:52"><i><font color="Green">E42</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>9</sub></font> = 0 as   <a href="turing_1.html#NM1">State</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E42"><i><font color="Green">Lemma31</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</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">E44</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#E5:52"><i><font color="Green">E41</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>11</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span>;<br/>
<i><font color="Green">E45</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#E1:52"><i><font color="Green">E40</font></i></a>, <a class="txt" href="turing_1.html#E7:52"><i><font color="Green">E43</font></i></a>, <a class="txt" href="turing_1.html#E47"><i><font color="Green">Lemma36</font></i></a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <font color="Maroon">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#E5:52"><i><font color="Green">E41</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>9</sub></font>,<font color="Maroon">c<sub>10</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#E1:52"><i><font color="Green">E40</font></i></a>, <a class="txt" href="turing_1.html#E10:52"><i><font color="Green">E44</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>12</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</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#K14" target="_self">SumTuring</a>  ;<br/>
<a NAME="E13:52"/><i><font color="Green">E46</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; <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"><span class="p3">(<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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">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 ) &amp; ( 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> <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> <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> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4 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="txt" href="turing_1.html#E1:52"><i><font color="Green">E40</font></i></a>, <a class="ref" href="turing_1.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E14:52"/><i><font color="Green">E47</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>10</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E15:52"/>
<font color="Maroon">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="E16:52"/><b>then </b><i><font color="Green">E48</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="E17:52"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>10</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="E18:52"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>10</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#E16:52"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E19:52"/><i><font color="Green">E51</font></i>: 
<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> <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/>
<b>set </b><font color="Maroon">c<sub>13</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/>
<a NAME="E20:52"/>
<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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E21:52"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 4 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E22:52"/><i><font color="Green">E53</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <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/>
<a NAME="E23:52"/><i><font color="Green">E54</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>10</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>10</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; <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>10</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"><span class="p3">(<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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">c<sub>10</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"><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>10</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 ) &amp; ( 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> <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> <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> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4 holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</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="52_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_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>10</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#E13:52"><i><font color="Green">E46</font></i></a>, <a class="txt" href="turing_1.html#E14:52"><i><font color="Green">E47</font></i></a>, <a class="ref" href="turing_1.html#T30" target="_self">Th30</a></i>;<br/>

<b>thus </b><a NAME="E2:52_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>10</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#E13:52"><i><font color="Green">E46</font></i></a>, <a class="txt" href="turing_1.html#E16:52"><i><font color="Green">E48</font></i></a>, <a class="txt" href="turing_1.html#E17:52"><i><font color="Green">E49</font></i></a>, <a class="ref" href="turing_1.html#T30" target="_self">Th30</a></i>;<br/>

<a NAME="E3:52_5"/>
<font color="Maroon">c<sub>10</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> 4
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E4:52_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>10</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"><span class="p3">(<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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="turing_1.html#E13:52"><i><font color="Green">E46</font></i></a>, <a class="txt" href="turing_1.html#E21:52"><i><font color="Green">E52</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>14</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>assume </b><a NAME="E1:52_5_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</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="E2:52_5_1"/><b>then </b><i><font color="Green">E56</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>14</sub></font>
 <b>by </b><i><a class="txt" href="turing_1.html#E14:52"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E1:52_5_1"><i><font color="Green">E55</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#E1:52"><i><font color="Green">E40</font></i></a>, <a class="txt" href="turing_1.html#E19:52"><i><font color="Green">E51</font></i></a>, <a class="txt" href="turing_1.html#E1:52_5_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="turing_1.html#E2:52_5_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="turing_1.html#T22" target="_self">Th22</a></i>
;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>assume </b><a NAME="E1:52_5_2"/><i><font color="Green">E55</font></i>: 
( <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> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</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> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4 )
 ;<br/><b>hence </b><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E18:52"><i><font color="Green">E50</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#E1:52"><i><font color="Green">E40</font></i></a>, <a class="txt" href="turing_1.html#E1:52_5_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="turing_1.html#T22" target="_self">Th22</a></i>
;<br/><br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<i><font color="Green">E55</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>9</sub></font>,<font color="Maroon">c<sub>10</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#K14" target="_self">SumTuring</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>9</sub></font>,<font color="Maroon">c<sub>10</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>12</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>9</sub></font>,<font color="Maroon">c<sub>10</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#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</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>12</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>9</sub></font>,<font color="Maroon">c<sub>10</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#D15" target="_self">Def15</a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</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>9</sub></font>,<font color="Maroon">c<sub>10</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#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</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>9</sub></font>,<font color="Maroon">c<sub>10</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#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</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">1,0,1</span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:52"><i><font color="Green">E40</font></i></a>, <a class="txt" href="turing_1.html#E14:52"><i><font color="Green">E47</font></i></a>, <a class="txt" href="turing_1.html#E18:52"><i><font color="Green">E50</font></i></a>, <a class="ref" href="turing_1.html#T17" target="_self">Th17</a>, <a class="ref" href="turing_1.html#T22" target="_self">Th22</a></i>
;<br/>
<i><font color="Green">E56</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>9</sub></font>,<font color="Maroon">c<sub>10</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>9</sub></font>,<font color="Maroon">c<sub>10</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>9</sub></font>,<font color="Maroon">c<sub>10</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>9</sub></font>,<font color="Maroon">c<sub>10</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>9</sub></font>,<font color="Maroon">c<sub>10</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>10</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>9</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E24:52"><i><font color="Green">E55</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/>
<i><font color="Green">E57</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>9</sub></font>,<font color="Maroon">c<sub>10</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>9</sub></font>,<font color="Maroon">c<sub>10</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#E24:52"><i><font color="Green">E55</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>14</sub></font> = 1 as   <a href="turing_1.html#NM1">State</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E42"><i><font color="Green">Lemma31</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>10</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>16</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</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>10</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">E58</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>9</sub></font>,<font color="Maroon">c<sub>10</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>9</sub></font>,<font color="Maroon">c<sub>10</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>9</sub></font>,<font color="Maroon">c<sub>10</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>9</sub></font>,<font color="Maroon">c<sub>10</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>10</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#E25:52"><i><font color="Green">E56</font></i></a>, <a class="txt" href="turing_1.html#E47"><i><font color="Green">Lemma36</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>9</sub></font>,<font color="Maroon">c<sub>10</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>9</sub></font>,<font color="Maroon">c<sub>10</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>10</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#E24:52"><i><font color="Green">E55</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>14</sub></font>,<font color="Maroon">c<sub>15</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>10</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#E26:52"><i><font color="Green">E57</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
;<br/>
<a NAME="E30:52"/><i><font color="Green">E59</font></i>: 
the <a href="turing_1.html#U3" target="_self">Tran</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</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>14</sub></font>,1,1</span><a href="domain_1.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="turing_1.html#D15" target="_self">Def15</a>, <a class="ref" href="turing_1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E31:52"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>14</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#K14" target="_self">SumTuring</a> 
 
<b>by </b><i><a class="ref" href="turing_1.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E32:52"/><i><font color="Green">E61</font></i>: 
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>10</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="52_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<b>assume </b><a NAME="E1:52_10"/><i><font color="Green">E62</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>17</sub></font> &amp; <font color="Maroon">c<sub>17</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="E2:52_10"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</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="E3:52_10"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="turing_1.html#E1:52_10"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E4:52_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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="turing_1.html#E23:52"><i><font color="Green">E54</font></i></a>, <a class="txt" href="turing_1.html#E1:52_10"><i><font color="Green">E62</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</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>18</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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>19</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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#K14" target="_self">SumTuring</a>  ;<br/>
<div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_11"><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>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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#K14" target="_self">SumTuring</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>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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>19</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>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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>19</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>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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#D15" target="_self">Def15</a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>19</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>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</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>10</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>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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#E23:52"><i><font color="Green">E54</font></i></a>, <a class="ref" href="turing_1.html#T17" target="_self">Th17</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>20</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font>;<br/>
<a NAME="E36:52"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>10</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="E37:52"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</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="E38:52"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</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#E36:52"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E39:52"/><i><font color="Green">E66</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</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> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">c<sub>10</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"><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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4 holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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="52_12"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="turing_1.html#E22:52"><i><font color="Green">E53</font></i></a>, <a class="txt" href="turing_1.html#E36:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="turing_1.html#E37:52"><i><font color="Green">E64</font></i></a>, <a class="ref" href="turing_1.html#T30" target="_self">Th30</a></i>;<br/>

<a NAME="E2:52_12"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</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> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E3:52_12"/><b>then </b><i><font color="Green">E67</font></i>: 
<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> <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> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="txt" href="turing_1.html#E19:52"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E4:52_12"/>
<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> <font color="Maroon">c<sub>5</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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E5:52_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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="turing_1.html#E23:52"><i><font color="Green">E54</font></i></a>, <a class="txt" href="turing_1.html#E3:52_12"><i><font color="Green">E67</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>21</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>assume </b><a NAME="E1:52_12_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</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> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4 )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_12_1_1"/>
( <font color="Maroon">c<sub>21</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> or <font color="Maroon">c<sub>21</sub></font> <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> or <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">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><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_12_1_1_1"><b>suppose </b></a><a NAME="E1:52_12_1_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>21</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/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</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#E23:52"><i><font color="Green">E54</font></i></a>, <a class="txt" href="turing_1.html#E1:52_12_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="turing_1.html#E1:52_12_1_1_1"><i><font color="Green">E69</font></i></a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_12_1_1_2"><b>suppose </b></a><a NAME="E1:52_12_1_1_2"/>
<font color="Maroon">c<sub>21</sub></font> <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/><div class="add"><b>hence </b><a NAME="E2:52_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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</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="52_12_1_1_3"><b>suppose </b></a><a NAME="E1:52_12_1_1_3"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">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/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</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#E23:52"><i><font color="Green">E54</font></i></a>, <a class="txt" href="turing_1.html#E1:52_12_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="turing_1.html#E1:52_12_1_1_3"><i><font color="Green">E69</font></i></a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>


</div><b>end;</b></div>
<i><font color="Green">E67</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>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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>10</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>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</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>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E35:52"><i><font color="Green">E62</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/>
<i><font color="Green">E68</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>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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#E35:52"><i><font color="Green">E62</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>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;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</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>23</sub></font> = 2 as   <a href="turing_1.html#NM1">State</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E42"><i><font color="Green">Lemma31</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>23</sub></font>,<font color="Maroon">c<sub>22</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>;<br/>
<i><font color="Green">E69</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>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E40:52"><i><font color="Green">E67</font></i></a>, <a class="txt" href="turing_1.html#E47"><i><font color="Green">Lemma36</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>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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>14</sub></font>,<font color="Maroon">c<sub>17</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>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E35:52"><i><font color="Green">E62</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>22</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E41:52"><i><font color="Green">E68</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
;<br/>
<a NAME="E45:52"/><i><font color="Green">E70</font></i>: 
the <a href="turing_1.html#U3" target="_self">Tran</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>23</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>23</sub></font>,1,1</span><a href="domain_1.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="turing_1.html#D15" target="_self">Def15</a>, <a class="ref" href="turing_1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E46:52"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>23</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#K14" target="_self">SumTuring</a> 
 
<b>by </b><i><a class="ref" href="turing_1.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E47:52"/>
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"><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 <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"><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> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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="52_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>25</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<b>assume </b><a NAME="E1:52_16"/><i><font color="Green">E72</font></i>: 
( <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 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/>

<a NAME="E2:52_16"/>
<font color="Maroon">c<sub>17</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="E3:52_16"/><b>then </b>
<font color="Maroon">c<sub>10</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="txt" href="turing_1.html#E38:52"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E4:52_16"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="turing_1.html#E1:52_16"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E5:52_16"/>
<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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="turing_1.html#E39:52"><i><font color="Green">E66</font></i></a>, <a class="txt" href="turing_1.html#E1:52_16"><i><font color="Green">E72</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E48:52"/><b>then </b><i><font color="Green">E72</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>22</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default">2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="turing_1.html#E45:52"><i><font color="Green">E70</font></i></a>, <a class="txt" href="turing_1.html#E46:52"><i><font color="Green">E71</font></i></a>, <a class="txt" href="turing_1.html#E49"><i><font color="Green">Lemma38</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>25</sub></font> = <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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4;<br/>
<b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">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>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>28</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>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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#K14" target="_self">SumTuring</a>  ;<br/>
<i><font color="Green">E73</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>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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#K14" target="_self">SumTuring</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>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>28</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>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>28</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>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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#D15" target="_self">Def15</a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>28</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>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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">3,0,<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#E39:52"><i><font color="Green">E66</font></i></a>, <a class="ref" href="turing_1.html#T17" target="_self">Th17</a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>29</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> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1;<br/>
<div><i><font color="Green">E74</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_18"><b>now </b></a><div class="add"><i><font color="Green">E75</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>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<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> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<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> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E51:52"><i><font color="Green">E73</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> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E39:52"><i><font color="Green">E66</font></i></a>, <a class="ref" href="turing_1.html#T28" target="_self">Th28</a></i>
;<br/><i><font color="Green">E76</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>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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#E51:52"><i><font color="Green">E73</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/><a NAME="E3:52_18"/><i><font color="Green">E77</font></i>: 
<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> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><b>thus </b> <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:52_18"><i><font color="Green">E75</font></i></a>, <a class="txt" href="turing_1.html#E47"><i><font color="Green">Lemma36</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>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>26</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E51:52"><i><font color="Green">E73</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">3,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E2:52_18"><i><font color="Green">E76</font></i></a>, <a class="txt" href="turing_1.html#E3:52_18"><i><font color="Green">E77</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>30</sub></font> = 3 as   <a href="turing_1.html#NM1">State</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E42"><i><font color="Green">Lemma31</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>31</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> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>32</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>33</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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#K14" target="_self">SumTuring</a>  ;<br/>
<a NAME="E56:52"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>10</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> 3
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E57:52"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="turing_1.html#E21:52"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E58:52"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>10</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> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="turing_1.html#E56:52"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E59:52"/><i><font color="Green">E78</font></i>: 
<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> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4
 
<b>by </b><i><a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/>
<a NAME="E60:52"/><b>then </b><i><font color="Green">E79</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="turing_1.html#E39:52"><i><font color="Green">E66</font></i></a>, <a class="txt" href="turing_1.html#E58:52"><i><font color="Green">E77</font></i></a></i>;<br/>
<div><i><font color="Green">E80</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_19"><b>now </b></a><div class="add"><b>thus </b> <a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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#K14" target="_self">SumTuring</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>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>33</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>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>33</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>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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#D15" target="_self">Def15</a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>33</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>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</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> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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,<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#E60:52"><i><font color="Green">E79</font></i></a>, <a class="ref" href="turing_1.html#T17" target="_self">Th17</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>34</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">c<sub>7</sub></font>;<br/>
<a NAME="E62:52"/><i><font color="Green">E81</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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"><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> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">c<sub>10</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"><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> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1 holds <br/><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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="52_20"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_20"/>
<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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="turing_1.html#E39:52"><i><font color="Green">E66</font></i></a>, <a class="txt" href="turing_1.html#E56:52"><i><font color="Green">E75</font></i></a>, <a class="txt" href="turing_1.html#E57:52"><i><font color="Green">E76</font></i></a>, <a class="ref" href="turing_1.html#T30" target="_self">Th30</a></i>;<br/>

<b>thus </b><a NAME="E2:52_20"/>
<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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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"><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> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><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>35</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>assume </b><a NAME="E1:52_20_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>35</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> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1 )
 ;<br/><a NAME="E2:52_20_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>35</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> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4
 <b>by </b><i><a class="txt" href="turing_1.html#E59:52"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E1:52_20_1"><i><font color="Green">E82</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#E39:52"><i><font color="Green">E66</font></i></a>, <a class="txt" href="turing_1.html#E1:52_20_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="turing_1.html#E2:52_20_1"><i><font color="Green">E83</font></i></a></i>
;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<i><font color="Green">E82</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>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E61:52"><i><font color="Green">E80</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/>
<i><font color="Green">E83</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>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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#E61:52"><i><font color="Green">E80</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>35</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1;<br/>
<a NAME="E65:52"/><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>36</sub></font> = 4 as   <a href="turing_1.html#NM1">State</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a>  <b>by </b><i><a class="txt" href="turing_1.html#E42"><i><font color="Green">Lemma31</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>37</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>38</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>;<br/>
<div><i><font color="Green">E85</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_23"><b>now </b></a><div class="add"><b>thus </b> <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#E63:52"><i><font color="Green">E82</font></i></a>, <a class="txt" href="turing_1.html#E47"><i><font color="Green">Lemma36</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>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#E61:52"><i><font color="Green">E80</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>36</sub></font>,<font color="Maroon">c<sub>37</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#E64:52"><i><font color="Green">E83</font></i></a>, <a class="txt" href="turing_1.html#E65:52"><i><font color="Green">E84</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
;<br/></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#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 implies <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default">4,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span> );<br/>
<a NAME="E69:52"/><i><font color="Green">E86</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="E70:52"/><i><font color="Green">E87</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="52_24"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>39</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:52_24"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>39</sub></font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_24_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_24_1"/><i><font color="Green">E89</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>39</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E2:52_24_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>39</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>39</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>40</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>39</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>41</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>39</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>42</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>43</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#K14" target="_self">SumTuring</a>  ;<br/><a NAME="E5:52_24_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>39</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 0
 <b>by </b><i><a class="txt" href="turing_1.html#E1:52_24_1"><i><font color="Green">E89</font></i></a></i>;<br/><a NAME="E6:52_24_1"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K6">-</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>39</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T23">XREAL_1:23</a></i>;<br/><a NAME="E7:52_24_1"/>
<font color="Maroon">c<sub>10</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="E8:52_24_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>39</sub></font>
 <b>by </b><i><a class="txt" href="turing_1.html#E6:52_24_1"><i><font color="Green">E91</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>44</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>39</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="E10:52_24_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>39</sub></font>
 <b>by </b><i><a class="ref" href="int_1.html#T19">INT_1:19</a></i>;<br/><a NAME="E11:52_24_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>39</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E12:52_24_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/><a NAME="E13:52_24_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>39</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> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="turing_1.html#E11:52_24_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E14:52_24_1"/><b>then </b><i><font color="Green">E93</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="turing_1.html#E62:52"><i><font color="Green">E81</font></i></a>, <a class="txt" href="turing_1.html#E6:52_24_1"><i><font color="Green">E91</font></i></a></i>;<br/><div><i><font color="Green">E94</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_24_1_1"><b>now </b></a><div class="add"><b>thus </b> <a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#K14" target="_self">SumTuring</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>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>43</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>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>43</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>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#D15" target="_self">Def15</a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>43</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>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>36</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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">4,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E14:52_24_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="turing_1.html#T17" target="_self">Th17</a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
;<br/></div><b>end;</b></div><div><i><font color="Green">E95</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_24_1_2"><b>now </b></a><div class="add"><b>thus </b> <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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> <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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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> <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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>39</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>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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> <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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>39</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E15:52_24_1"><i><font color="Green">E94</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E14:52_24_1"><i><font color="Green">E93</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">E96</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>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#E15:52_24_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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="p1">(<span class="default"><font color="Maroon">c<sub>39</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>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#E1:52_24"><i><font color="Green">E88</font></i></a>, <a class="txt" href="turing_1.html#E1:52_24_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="turing_1.html#E2:52_24_1"><i><font color="Green">E90</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>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#E16:52_24_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="turing_1.html#E47"><i><font color="Green">Lemma36</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>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>36</sub></font>,<font color="Maroon">c<sub>41</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#E15:52_24_1"><i><font color="Green">E94</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>39</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#E17:52_24_1"><i><font color="Green">E96</font></i></a>, <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">4,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>39</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:52_24"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>39</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>39</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<a NAME="E71:52"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E72:52"/><b>then </b><i><font color="Green">E88</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"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>40</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>;<br/>
<a NAME="E73:52"/>
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#E69:52"><i><font color="Green">E86</font></i></a>, <a class="txt" href="turing_1.html#E70:52"><i><font color="Green">E87</font></i></a>);<br/></i>
<b>then </b><i><font color="Green">E89</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>36</sub></font>,<font color="Maroon">c<sub>37</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default">4,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#E72:52"><i><font color="Green">E88</font></i></a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span>

;<br/>
<div><i><font color="Green">E90</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_26"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon">c<sub>41</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#K14" target="_self">SumTuring</a>  ;<br/><i><font color="Green">E91</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>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#K14" target="_self">SumTuring</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>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>41</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>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>41</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>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#D15" target="_self">Def15</a></i>
<br/>.= 
<a href="turing_1.html#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">4,<span class="p2">(<span class="default"><font color="Maroon">c<sub>41</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>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#K13" target="_self">Sum_Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">4,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p5">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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">5,0,0</span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E62:52"><i><font color="Green">E81</font></i></a>, <a class="ref" href="turing_1.html#T17" target="_self">Th17</a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
;<br/><i><font color="Green">E92</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>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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> <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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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> <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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>10</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>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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> <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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E2:52_26"><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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E62:52"><i><font color="Green">E81</font></i></a>, <a class="ref" href="turing_1.html#T28" target="_self">Th28</a></i>
;<br/><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>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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/>.= 
0
<b>by </b><i><a class="txt" href="turing_1.html#E2:52_26"><i><font color="Green">E91</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>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#E3:52_26"><i><font color="Green">E92</font></i></a>, <a class="txt" href="turing_1.html#E47"><i><font color="Green">Lemma36</font></i></a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default">5,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>36</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#E2:52_26"><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">5,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K2">+</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p3">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p4">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#E4:52_26"><i><font color="Green">E93</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>41</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/>
<span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</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>10</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#E11:52"><i><font color="Green">E45</font></i></a>, <a class="txt" href="turing_1.html#E29:52"><i><font color="Green">E58</font></i></a>, <a class="ref" href="turing_1.html#T12" target="_self">Th12</a></i>
;<br/>
<a NAME="E77:52"/><b>then </b>
<span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">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"><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> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</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>10</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="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self">Th13</a></i>;<br/>
<b>then </b><i><font color="Green">E91</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"><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> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
 <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</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>10</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="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> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self">Def8</a></i>
<br/>.= 
 <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1">(<span class="default"><a href="turing_1.html#K9" target="_self">Following</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</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>10</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> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self">Def8</a></i>
<br/>.= 
 <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1">(<span class="default"><a href="turing_1.html#K9" target="_self">Following</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</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>10</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="p5">(<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> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self">Th13</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_29"><b>now </b></a><div class="add"><span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2">(<span class="default"><a href="turing_1.html#K9" target="_self">Following</a> <span class="p3">(<span class="default"><a href="turing_1.html#K9" target="_self">Following</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</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>10</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="p1">(<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> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E78:52"><i><font color="Green">E91</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"><a href="turing_1.html#K9" target="_self">Following</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0">(<span class="default"><a href="turing_1.html#K9" target="_self">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</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>10</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></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="turing_1.html#D8" target="_self">Def8</a></i>
;<br/><b>hence </b><a NAME="E2:52_29"/>
<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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2">(<span class="default"><a href="turing_1.html#K9" target="_self">Following</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>31</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="domain_1.html#K4">]</a></span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="turing_1.html#E30:52"><i><font color="Green">E59</font></i></a>, <a class="txt" href="turing_1.html#E31:52"><i><font color="Green">E60</font></i></a>, <a class="txt" href="turing_1.html#E32:52"><i><font color="Green">E61</font></i></a>, <a class="txt" href="turing_1.html#E44:52"><i><font color="Green">E69</font></i></a>, <a class="txt" href="turing_1.html#E48:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="turing_1.html#E52:52"><i><font color="Green">E74</font></i></a>, <a class="txt" href="turing_1.html#E49"><i><font color="Green">Lemma38</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E80:52"/><b>then </b><i><font color="Green">E92</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default">5,<font color="Maroon">c<sub>10</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">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#E68:52"><i><font color="Green">E85</font></i></a>, <a class="txt" href="turing_1.html#E74:52"><i><font color="Green">E89</font></i></a>, <a class="txt" href="turing_1.html#E75:52"><i><font color="Green">E90</font></i></a>, <a class="ref" href="turing_1.html#D8" target="_self">Def8</a></i>;<br/>
<b>then </b><i><font color="Green">E93</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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a>  = 
5
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a> 
<b>by </b><i><a class="ref" href="turing_1.html#D15" target="_self">Def15</a></i>
;<br/>
<b>hence </b><a NAME="E82:52"/>
<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="E83:52"/><b>then </b><i><font color="Green">E94</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="turing_1.html#E81:52"><i><font color="Green">E93</font></i></a>, <a class="ref" href="turing_1.html#D10" target="_self">Def10</a></i>;<br/>
<b>hence </b><a NAME="E84:52"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="turing_1.html#E80:52"><i><font color="Green">E92</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/>

<a NAME="E85:52"/><i><font color="Green">E95</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>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="turing_1.html#E80:52"><i><font color="Green">E92</font></i></a>, <a class="txt" href="turing_1.html#E83:52"><i><font color="Green">E94</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/>
<a NAME="E86:52"/>
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p2">(<span class="default"><a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">c<sub>7</sub></font> <a href="turing_1.html#R1" target="_self">is_1_between</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 
<b>by </b><i><a class="txt" href="turing_1.html#E62:52"><i><font color="Green">E81</font></i></a>, <a class="ref" href="turing_1.html#D13" target="_self">Def13</a></i>;<br/>
<b>hence </b><a NAME="E87:52"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">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"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="turing_1.html#E85:52"><i><font color="Green">E95</font></i></a>, <a class="ref" href="turing_1.html#T19" target="_self">Th19</a></i>;<br/>


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