<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">T</font> be    <a href="turing_1.html#L1" title="TURING_1:struct.1">TuringStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">t</font> being   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <font color="Maroon" title="c1">T</font><br/>  for <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">n</font> being    <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>   st <font color="Olive" title="b1">t</font> <a href="turing_1.html#R1" title="TURING_1:pred.1">is_1_between</a> <font color="Olive" title="b2">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">s</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 holds <br/><font color="Olive" title="b1">t</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b2">s</font>,<font color="Olive" title="b3">n</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">n</font> being    <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>   st <font color="Maroon" title="c2">t</font> <a href="turing_1.html#R1" title="TURING_1:pred.1">is_1_between</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 holds <br/><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#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">n</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">n</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 implies <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#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c5">f</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>;<br/>
<span class="kw">assume </span><a NAME="E1:35"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<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#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span><br/>

<a NAME="E2:35"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<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#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</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#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#T4" target="_self" title="TURING_1:th.4">Th7</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">i</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> implies <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"><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#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Maroon" title="c6">i</font>)</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>,<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#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">i</font></span>)</span> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">and </span><br/><a NAME="E2:35_1"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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"><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#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Maroon" title="c6">i</font>)</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>,<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#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">i</font></span>)</span></span><br/><a NAME="E3:35_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><a NAME="E4:35_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="int_1.html#T7" title="INT_1:th.7">INT_1:7</a></span>;<br/><a NAME="E5:35_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E6:35_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:35_1"/>
<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"><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#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Maroon" title="c6">i</font>)</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>,<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#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="turing_1.html#T4" target="_self" title="TURING_1:th.4">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:35"/>
<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#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">n</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="turing_1.html#D13" target="_self" title="TURING_1:def.13">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
