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<b>let </b><font color="Maroon" title="c1">T</font> be    <a href="turing_1.html#L1" title="TURING_1:struct.1">TuringStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">t</font> be   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">n1</font>, <font color="Maroon" title="c5">n2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>





<b>set </b><font color="Maroon" title="c6">f</font> = <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n1</font>,<font color="Maroon" title="c5">n2</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>;<br/>
<b>assume </b><a NAME="E1:36"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c2">t</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n1</font>,<font color="Maroon" title="c5">n2</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 ;<br/>

<a NAME="E2:36"/><i><font color="Green" title="E24">A2</font></i>: 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p2">(<span class="default"><a href="turing_1.html#K4" title="TURING_1:func.4">Prefix</a> <span class="p3"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n1</font>,<font color="Maroon" title="c5">n2</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font>
 
<b>by </b><i><a class="ref" href="turing_1.html#T8" target="_self" title="TURING_1:th.8">Th8</a></i>;<br/>
<a NAME="E3:36"/><i><font color="Green" title="E25">A3</font></i>: 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p2">(<span class="default"><a href="turing_1.html#K4" title="TURING_1:func.4">Prefix</a> <span class="p3"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n1</font>,<font color="Maroon" title="c5">n2</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<b>by </b><i><a class="ref" href="turing_1.html#T8" target="_self" title="TURING_1:th.8">Th8</a></i>;<br/>
<a NAME="E4:36"/><i><font color="Green" title="E26">A4</font></i>: 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p2">(<span class="default"><a href="turing_1.html#K4" title="TURING_1:func.4">Prefix</a> <span class="p3"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n1</font>,<font color="Maroon" title="c5">n2</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<b>by </b><i><a class="ref" href="turing_1.html#T8" target="_self" title="TURING_1:th.8">Th8</a></i>;<br/>
<a NAME="E5:36"/><i><font color="Green" title="E27">A5</font></i>: 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p2">(<span class="default"><a href="turing_1.html#K4" title="TURING_1:func.4">Prefix</a> <span class="p3"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n1</font>,<font color="Maroon" title="c5">n2</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4
 
<b>by </b><i><a class="ref" href="turing_1.html#T8" target="_self" title="TURING_1:th.8">Th8</a></i>;<br/>
<a NAME="E6:36"/><i><font color="Green" title="E28">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n1</font>,<font color="Maroon" title="c5">n2</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>
<b>hence </b><a NAME="E7:36"/>
<font color="Maroon" title="c2">t</font> <a href="turing_1.html#R1" title="TURING_1:pred.1">is_1_between</a> <font color="Maroon" title="c3">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <b>by </b><i><a class="txt" href="turing_1.html#E1:36"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="turing_1.html#E2:36"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="turing_1.html#E3:36"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="turing_1.html#D14" target="_self" title="TURING_1:def.14">Def14</a></i>;<br/>

<b>thus </b><a NAME="E8:36"/>
<font color="Maroon" title="c2">t</font> <a href="turing_1.html#R1" title="TURING_1:pred.1">is_1_between</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4
 <b>by </b><i><a class="txt" href="turing_1.html#E1:36"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="turing_1.html#E4:36"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="turing_1.html#E5:36"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="turing_1.html#E6:36"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="turing_1.html#D14" target="_self" title="TURING_1:def.14">Def14</a></i>;<br/>


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