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<b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">tm1</font> be    <a href="turing_1.html#L1" target="_self">TuringStr</a> , <font color="Maroon">tm2</font> be    <a href="turing_1.html#L1" target="_self">TuringStr</a> ;<br/><b>let </b><font color="Maroon">t1</font> be   <a href="turing_1.html#NM2">Tape</a> of <font color="Maroon">tm1</font>;<br/><b>let </b><font color="Maroon">t2</font> be   <a href="turing_1.html#NM2">Tape</a> of <font color="Maroon">tm2</font>;<br/>







<b>assume </b><a NAME="E1:100"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> &amp; <font color="Maroon">t1</font> <a href="turing_1.html#R2" target="_self">storeData</a> <font color="Maroon">f</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:100_1"/><i><font color="Green">E31</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 ;<br/><b>set </b><font color="Maroon">m</font> = <span class="p1">(<span class="default"><a href="gr_cy_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="turing_1.html#K4" target="_self">Prefix</a> <font color="Maroon">f</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">n</font> = <span class="p1">(<span class="default"><a href="gr_cy_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="turing_1.html#K4" target="_self">Prefix</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span>;<br/><a NAME="E2:100_1"/>
<font color="Maroon">t1</font> <a href="turing_1.html#R1" target="_self">is_1_between</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="turing_1.html#K4" target="_self">Prefix</a> <font color="Maroon">f</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="turing_1.html#K4" target="_self">Prefix</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="turing_1.html#T14" target="_self">Th14</a>, , </i>;<br/><a NAME="E3:100_1"/><b>then </b>
( <font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_1.html#K2">Sum</a> <span class="p3">(<span class="default"><a href="turing_1.html#K4" target="_self">Prefix</a> <font color="Maroon">f</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_1.html#K2">Sum</a> <span class="p3">(<span class="default"><a href="turing_1.html#K4" target="_self">Prefix</a> <font color="Maroon">f</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">k</font> being  <a href="int_1.html#NM1">Integer</a>  st <span class="p1">(<span class="default"><a href="gr_cy_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="turing_1.html#K4" target="_self">Prefix</a> <font color="Maroon">f</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="turing_1.html#K4" target="_self">Prefix</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> holds <br/><font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 1 ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:100_1"/>
<font color="Maroon">t2</font> <a href="turing_1.html#R1" target="_self">is_1_between</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="turing_1.html#K4" target="_self">Prefix</a> <font color="Maroon">f</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="turing_1.html#K4" target="_self">Prefix</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="turing_1.html#T14" target="_self">Th14</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:100"/>
<font color="Maroon">t2</font> <a href="turing_1.html#R2" target="_self">storeData</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>


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