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<b>let </b><font color="Maroon" title="c1">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">tm1</font>, <font color="Olive" title="b2">tm2</font> being   <a href="turing_1.html#L1" title="TURING_1:struct.1">TuringStr</a> <br/> for <font color="Olive" title="b3">t1</font> being  <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <font color="Olive" title="b1">tm1</font><br/> for <font color="Olive" title="b4">t2</font> being  <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <font color="Olive" title="b2">tm2</font>  st <font color="Olive" title="b3">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">t2</font> &amp; <font color="Olive" title="b3">t1</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <font color="Maroon" title="c1">f</font> holds <br/><font color="Olive" title="b4">t2</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <font color="Maroon" title="c1">f</font></span><br/><b>let </b><font color="Maroon" title="c2">tm1</font>, <font color="Maroon" title="c3">tm2</font> be    <a href="turing_1.html#L1" title="TURING_1:struct.1">TuringStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">t1</font> being  <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <font color="Maroon" title="c2">tm1</font><br/> for <font color="Olive" title="b2">t2</font> being  <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <font color="Maroon" title="c3">tm2</font>  st <font color="Olive" title="b1">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t2</font> &amp; <font color="Olive" title="b1">t1</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <font color="Maroon" title="c1">f</font> holds <br/><font color="Olive" title="b2">t2</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <font color="Maroon" title="c1">f</font></span><br/><b>let </b><font color="Maroon" title="c4">t1</font> be   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <font color="Maroon" title="c2">tm1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">t2</font> being  <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <font color="Maroon" title="c3">tm2</font>  st <font color="Maroon" title="c4">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t2</font> &amp; <font color="Maroon" title="c4">t1</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <font color="Maroon" title="c1">f</font> holds <br/><font color="Olive" title="b1">t2</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <font color="Maroon" title="c1">f</font></span><br/><b>let </b><font color="Maroon" title="c5">t2</font> be   <a href="turing_1.html#NM2" title="TURING_1:NM.2">Tape</a> of <font color="Maroon" title="c3">tm2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t2</font> &amp; <font color="Maroon" title="c4">t1</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <font color="Maroon" title="c1">f</font> implies <font color="Maroon" title="c5">t2</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <font color="Maroon" title="c1">f</font> )</span><br/>







<b>assume </b><a NAME="E1:98"/><i><font color="Green" title="E71">A1</font></i>: 
( <font color="Maroon" title="c4">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t2</font> &amp; <font color="Maroon" title="c4">t1</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <font color="Maroon" title="c1">f</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">t2</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <font color="Maroon" title="c1">f</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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> <font color="Maroon" title="c1">f</font> implies <font color="Maroon" title="c5">t2</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> <font color="Maroon" title="c1">f</font>,<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> <font color="Maroon" title="c1">f</font>,<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/><b>assume </b><a NAME="E1:98_1"/><i><font color="Green" title="E72">A2</font></i>: 
( 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> <font color="Maroon" title="c1">f</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">t2</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> <font color="Maroon" title="c1">f</font>,<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> <font color="Maroon" title="c1">f</font>,<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/><b>set </b><font color="Maroon" title="c7">m</font> = <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> <font color="Maroon" title="c1">f</font>,<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>;<br/><b>set </b><font color="Maroon" title="c8">n</font> = <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> <font color="Maroon" title="c1">f</font>,<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>;<br/><a NAME="E2:98_1"/>
<font color="Maroon" title="c4">t1</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> <font color="Maroon" title="c1">f</font>,<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> <font color="Maroon" title="c1">f</font>,<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>
 <b>by </b><i><a class="txt" href="#E1:98"><i><font color="Green" title="E71">A1</font></i></a>, <a class="txt" href="#E1:98_1"><i><font color="Green" title="E72">A2</font></i></a>, <a class="ref" href="turing_1.html#D14" target="_self" title="TURING_1:def.14">Def14</a></i>;<br/><a NAME="E3:98_1"/><b>then </b>
( <font color="Maroon" title="c4">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p3">(<span class="default"><a href="turing_1.html#K4" title="TURING_1:func.4">Prefix</a> <font color="Maroon" title="c1">f</font>,<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="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p3">(<span class="default"><a href="turing_1.html#K4" title="TURING_1:func.4">Prefix</a> <font color="Maroon" title="c1">f</font>,<span class="p4">(<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="p2">(<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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <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> <font color="Maroon" title="c1">f</font>,<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> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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> <font color="Maroon" title="c1">f</font>,<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> holds <br/><font color="Maroon" title="c4">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 <b>by </b><i><a class="ref" href="turing_1.html#D13" target="_self" title="TURING_1:def.13">Def13</a></i>;<br/><b>hence </b><a NAME="E4:98_1"/>
<font color="Maroon" title="c5">t2</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> <font color="Maroon" title="c1">f</font>,<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> <font color="Maroon" title="c1">f</font>,<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>
 <b>by </b><i><a class="txt" href="#E1:98"><i><font color="Green" title="E71">A1</font></i></a>, <a class="ref" href="turing_1.html#D13" target="_self" title="TURING_1:def.13">Def13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:98"/>
<font color="Maroon" title="c5">t2</font> <a href="turing_1.html#R2" title="TURING_1:pred.2">storeData</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="ref" href="turing_1.html#D14" target="_self" title="TURING_1:def.14">Def14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
