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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">k</font>, <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">ie</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">ie</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> $1,<font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b3">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>for <font color="Olive" title="b5">e</font> being  <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of $1, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <font color="Olive" title="b4">ie</font>,$1,<font color="Olive" title="b2">k</font> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Olive" title="b1">m</font>,<font color="Olive" title="b5">e</font>,<font color="Olive" title="b3">f</font>,<font color="Olive" title="b2">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b4">ie</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Olive" title="b3">f</font>;<br/>
<a NAME="E1:26"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:26_1"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">and </span><br/><a NAME="E2:26_1"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">k</font>, <font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">ie</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">ie</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies for <font color="Olive" title="b1">e</font> being  <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <font color="Maroon" title="c5">ie</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">ie</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font> )</span><br/>

<a NAME="E3:26_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:26_1"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c5">ie</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c3">k</font>
 <span class="kw">and </span><br/><a NAME="E5:26_1"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">e</font> being  <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <font color="Maroon" title="c5">ie</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">ie</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">e</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <font color="Maroon" title="c5">ie</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font> implies <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">ie</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E6:26_1"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <font color="Maroon" title="c5">ie</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">ie</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">n</font> = <font color="Maroon" title="c1">n</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <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>  = <font color="Maroon" title="c6">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">en</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E9:26_1"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">en</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">and </span><br/><a NAME="E10:26_1"/><span class="lab"><font color="Green" title="E26">A9</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">en</font> holds <br/><font color="Maroon" title="c8">en</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></span>
<a NAME="E11:26_1"/><span class="lab"><font color="Green" title="E27">A10</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">en</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">en</font> = <font color="Maroon" title="c8">en</font> as   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c7">n</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E13:26_1"/><span class="lab"><font color="Green" title="E28">A11</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E14:26_1"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
<font color="Maroon" title="c5">ie</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="radix_1.html#D12" title="RADIX_1:def.12">RADIX_1:def 12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">ien</font> =  <a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font>;<br/>
<a NAME="E15:26_1"/><span class="lab"><font color="Green" title="E30">A13</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E16:26_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A14</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></span>;<br/>
<a NAME="E17:26_1"/><span class="lab"><font color="Green" title="E32">A15</font></span>: 
<font color="Maroon" title="c5">ie</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:26_1_1"/><span class="lab"><font color="Green" title="E33">A16</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <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>  = <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> $1;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">r2</font> =  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">rD</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E4:26_1_1"/><span class="lab"><font color="Green" title="E34">A17</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">rD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">and </span><br/><a NAME="E5:26_1_1"/><span class="lab"><font color="Green" title="E35">A18</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">rD</font> holds <br/><font color="Maroon" title="c12">rD</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c13">rD</font> = <font color="Maroon" title="c12">rD</font> as   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c7">n</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E7:26_1_1"/><span class="lab"><font color="Green" title="E36">A19</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">rD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">rD1</font> = <font color="Maroon" title="c13">rD</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T151" title="FINSEQ_2:th.151">FINSEQ_2:151</a></span>;<br/>
<span class="lab"><font color="Green" title="E37">A20</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E38">A21</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r2</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c14">rD1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_1"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="txt" href="#E7:26_1_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>
<br/>.= 
<font color="Maroon" title="c7">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>
;<br/>
<span class="lab"><font color="Green" title="E39">A22</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<br/>.= 
<font color="Maroon" title="c7">n</font> <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="#E10:26_1_1"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>
;<br/>
<a NAME="E12:26_1_1"/><span class="lab"><font color="Green" title="E40">A23</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E13:26_1_1"/><span class="lab"><font color="Green" title="E41">A24</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c15">i</font> &amp; <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> implies <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:26_1_1_4"/><span class="lab"><font color="Green" title="E42">A25</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font>
 <span class="kw">and </span><br/><a NAME="E2:26_1_1_4"/><span class="lab"><font color="Green" title="E43">A26</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span><br/>

<a NAME="E3:26_1_1_4"/><span class="lab"><font color="Green" title="E44">A27</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <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="#E2:26_1_1_4"><span class="lab"><font color="Green" title="E43">A26</font></span></a>, <a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E4:26_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A28</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_4"><span class="lab"><font color="Green" title="E42">A25</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E5:26_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A29</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26_1_1"><span class="lab"><font color="Green" title="E40">A23</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_1_4_1"/>
( <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c15">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_4_1_1"/><span class="lab"><font color="Green" title="E47">A30</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> = 
 <a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T58" title="FINSEQ_1:th.58">FINSEQ_1:58</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:26_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="ref" href="radix_2.html#D2" target="_self" title="RADIX_2:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_1_4"><span class="lab"><font color="Green" title="E46">A29</font></span></a>, <a class="txt" href="#E1:26_1_1_4_1_1"><span class="lab"><font color="Green" title="E47">A30</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:26_1_1_4_1_1"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_4_1_1"><span class="lab"><font color="Green" title="E47">A30</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_4_1_2"/><span class="lab"><font color="Green" title="E47">A31</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c16">rs2</font> = <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> as    <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/><span class="kw">reconsider </span><font color="Maroon" title="c17">rs</font> = <font color="Maroon" title="c13">rD</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E4:26_1_1_4_1_2"/>
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_4"><span class="lab"><font color="Green" title="E42">A25</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">i1</font> = <font color="Maroon" title="c15">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/><a NAME="E6:26_1_1_4_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_4"><span class="lab"><font color="Green" title="E42">A25</font></span></a>, <a class="txt" href="#E1:26_1_1_4_1_2"><span class="lab"><font color="Green" title="E47">A31</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E7:26_1_1_4_1_2"/><span class="kw">then </span>
1 <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> <font color="Maroon" title="c15">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></span>;<br/><a NAME="E8:26_1_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A32</font></span>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E9:26_1_1_4_1_2"/><span class="lab"><font color="Green" title="E49">A33</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1_4"><span class="lab"><font color="Green" title="E44">A27</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E10:26_1_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A34</font></span>: 
<font color="Maroon" title="c18">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26_1_1_4_1_2"><span class="lab"><font color="Green" title="E48">A32</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E11:26_1_1_4_1_2"/><span class="lab"><font color="Green" title="E51">A35</font></span>: 
<font color="Maroon" title="c18">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">rD</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_1_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E8:26_1_1_4_1_2"><span class="lab"><font color="Green" title="E48">A32</font></span></a>, <a class="txt" href="#E9:26_1_1_4_1_2"><span class="lab"><font color="Green" title="E49">A33</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E12:26_1_1_4_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_4"><span class="lab"><font color="Green" title="E42">A25</font></span></a>, <a class="txt" href="#E1:26_1_1_4_1_2"><span class="lab"><font color="Green" title="E47">A31</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">then </span><span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E11:26_1_1"><span class="lab"><font color="Green" title="E39">A22</font></span></a>, <a class="txt" href="#E3:26_1_1_4"><span class="lab"><font color="Green" title="E44">A27</font></span></a>, <a class="ref" href="finseq_1.html#T37" title="FINSEQ_1:th.37">FINSEQ_1:37</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">r2</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c13">rD</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_1"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="txt" href="#E9:26_1_1"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="txt" href="#E7:26_1_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>
<br/>.= 
<font color="Maroon" title="c11">r2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c17">rs</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T67" title="RVSUM_1:th.67">RVSUM_1:67</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c16">rs2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_1"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="txt" href="#E9:26_1_1"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="txt" href="#E7:26_1_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26_1_1"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="txt" href="#E7:26_1_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E11:26_1_1_4_1_2"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c18">i1</font>,<font color="Maroon" title="c3">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26_1_1_4_1_2"><span class="lab"><font color="Green" title="E50">A34</font></span></a>, <a class="ref" href="radix_2.html#D2" target="_self" title="RADIX_2:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">en</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">i1</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">en</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">i1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c18">i1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">en</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">i1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26_1_1_4_1_2"><span class="lab"><font color="Green" title="E48">A32</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c18">i1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E11:26_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E10:26_1_1_4_1_2"><span class="lab"><font color="Green" title="E50">A34</font></span></a></span>
<br/>.= 
 <a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_4"><span class="lab"><font color="Green" title="E42">A25</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_4"><span class="lab"><font color="Green" title="E45">A28</font></span></a>, <a class="ref" href="radix_2.html#D2" target="_self" title="RADIX_2:def.2">Def2</a></span>
;<br/><span class="kw">hence </span><a NAME="E14:26_1_1_4_1_2"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_1_4"><span class="lab"><font color="Green" title="E46">A29</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E14:26_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:26_1_1"><span class="lab"><font color="Green" title="E39">A22</font></span></a>, <a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E15:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A36</font></span>: 
 <a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:26_1_1"><span class="lab"><font color="Green" title="E41">A24</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/>
<a NAME="E16:26_1_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">e</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E17:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A37</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:26_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E18:26_1_1"/><span class="lab"><font color="Green" title="E44">A38</font></span>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<font color="Maroon" title="c5">ie</font> = 
 <a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E6:26_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="radix_2.html#T15" target="_self" title="RADIX_2:th.15">Th15</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:26_1_1"><span class="lab"><font color="Green" title="E42">A36</font></span></a>, <a class="ref" href="rvsum_1.html#T106" title="RVSUM_1:th.106">RVSUM_1:106</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">r2</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c13">rD</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_1"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="txt" href="#E9:26_1_1"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="txt" href="#E7:26_1_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c13">rD</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T117" title="RVSUM_1:th.117">RVSUM_1:117</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_1"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="txt" href="#E9:26_1_1"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="txt" href="#E7:26_1_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:26_1_1"><span class="lab"><font color="Green" title="E44">A38</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E20:26_1_1"/>
<font color="Maroon" title="c5">ie</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:26_1_1"><span class="lab"><font color="Green" title="E43">A37</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E18:26_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">ie</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T19" title="INT_1:th.19">INT_1:19</a></span>;<br/>
<a NAME="E19:26_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E20:26_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></span>;<br/>
<a NAME="E21:26_1"/><span class="kw">then </span>
 <a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c7">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/>
<a NAME="E22:26_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A39</font></span>: 
 <a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_1.html#D12" title="RADIX_1:def.12">RADIX_1:def 12</a></span>;<br/>
<a NAME="E23:26_1"/><span class="lab"><font color="Green" title="E34">A40</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E24:26_1"/><span class="lab"><font color="Green" title="E35">A41</font></span>: 
for <font color="Olive" title="b1">i</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">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <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> ] <span class="kw">means </span>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font> implies <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> $1 );<br/>
<a NAME="E1:26_1_2"/><span class="lab"><font color="Green" title="E36">A42</font></span>: 
for <font color="Olive" title="b1">i</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">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">i</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:26_1_2_1"/><span class="lab"><font color="Green" title="E37">A43</font></span>: 
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font> implies <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E2:26_1_2_1"/><span class="lab"><font color="Green" title="E38">A44</font></span>: 
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E3:26_1_2_1"/><span class="lab"><font color="Green" title="E39">A45</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">en</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:26_1_2_1"/><span class="lab"><font color="Green" title="E40">A46</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <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"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<a NAME="E5:26_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A47</font></span>: 
<font color="Maroon" title="c11">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> <font color="Maroon" title="c7">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E6:26_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A48</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E7:26_1_2_1"/><span class="lab"><font color="Green" title="E43">A49</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">e</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E8:26_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A50</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_2_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_2_1_1_1"/>
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_2_1"><span class="lab"><font color="Green" title="E38">A44</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:26_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A51</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <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"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
 <a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c7">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="radix_2.html#D9" target="_self" title="RADIX_2:def.9">Def9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">en</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:26_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E3:26_1_2_1"><span class="lab"><font color="Green" title="E39">A45</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E11:26_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E13:26_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a></span>
<br/>.= 
 <a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:26_1"><span class="lab"><font color="Green" title="E34">A40</font></span></a>, <a class="txt" href="#E8:26_1_2_1"><span class="lab"><font color="Green" title="E44">A50</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="radix_2.html#D9" target="_self" title="RADIX_2:def.9">Def9</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:26_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <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"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E45">A51</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:26_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A52</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <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"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c12">nn</font> = <font color="Maroon" title="c7">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 as    <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></span>;<br/><a NAME="E3:26_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E46">A53</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">nn</font> <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="#E6:26_1_2_1"><span class="lab"><font color="Green" title="E42">A48</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E4:26_1_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 ;<br/><a NAME="E5:26_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A54</font></span>: 
 ex <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">I1</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E1:26_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="ref" href="radix_2.html#D9" target="_self" title="RADIX_2:def.9">Def9</a></span>;<br/><a NAME="E6:26_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E48">A55</font></span>: 
 ex <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">I1</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E6:26_1_2_1"><span class="lab"><font color="Green" title="E42">A48</font></span></a>, <a class="txt" href="#E1:26_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="ref" href="radix_2.html#D9" target="_self" title="RADIX_2:def.9">Def9</a></span>;<br/><a NAME="E7:26_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E49">A56</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T35" title="NAT_D:th.35">NAT_D:35</a></span>;<br/><a NAME="E8:26_1_2_1_1_1_2"/>
<font color="Maroon" title="c11">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> <font color="Maroon" title="c7">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_2_1"><span class="lab"><font color="Green" title="E40">A46</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E9:26_1_2_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E10:26_1_2_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c12">nn</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T41" title="NAT_D:th.41">NAT_D:41</a></span>;<br/><a NAME="E11:26_1_2_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><a NAME="E12:26_1_2_1_1_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T35" title="NAT_D:th.35">NAT_D:35</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:26_1_2_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E14:26_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A57</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_1_2_1"><span class="lab"><font color="Green" title="E43">A49</font></span></a>, <a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/><a NAME="E15:26_1_2_1_1_1_2"/>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_2_1"><span class="lab"><font color="Green" title="E41">A47</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E16:26_1_2_1_1_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A53</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E17:26_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A58</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A56</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E18:26_1_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">en</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">en</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_2_1"><span class="lab"><font color="Green" title="E37">A43</font></span></a>, <a class="txt" href="#E1:26_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="txt" href="#E3:26_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A53</font></span></a>, <a class="txt" href="#E6:26_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E48">A55</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<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"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E11:26_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E17:26_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E51">A58</font></span></a></span>
<br/>.= 
<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"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A53</font></span></a>, <a class="ref" href="nat_d.html#T38" title="NAT_D:th.38">NAT_D:38</a></span>
<br/>.= 
<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"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E50">A57</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
;<br/><span class="kw">hence </span><a NAME="E20:26_1_2_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <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"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E47">A54</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:26_1_2_1"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <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"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:26_1_2"/><span class="lab"><font color="Green" title="E37">A59</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E3:26_1_2"/>
for <font color="Olive" title="b1">i</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>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:26_1_2"><span class="lab"><font color="Green" title="E37">A59</font></span></a>, <a class="txt" href="#E1:26_1_2"><span class="lab"><font color="Green" title="E36">A42</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:26_1_2"/>
for <font color="Olive" title="b1">i</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">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E25:26_1"/><span class="lab"><font color="Green" title="E36">A60</font></span>: 
 <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_2.html#T6" target="_self" title="RADIX_2:th.6">Th6</a></span>;<br/>
<a NAME="E26:26_1"/><span class="lab"><font color="Green" title="E37">A61</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">en</font> holds <br/><font color="Maroon" title="c9">en</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:26_1_3"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">e</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E2:26_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A62</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<span class="lab"><font color="Green" title="E39">A63</font></span>: <font color="Maroon" title="c6">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 = 
 <a href="radix_2.html#K4" title="RADIX_2:func.4">DigitDC2</a> <font color="Maroon" title="c5">ie</font>,1,<font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E16:26_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="ref" href="radix_2.html#D5" target="_self" title="RADIX_2:def.5">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">ie</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">ie</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></span>
<br/>.= 
<font color="Maroon" title="c5">ie</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T6" title="NAT_2:th.6">NAT_2:6</a></span>
<br/>.= 
<font color="Maroon" title="c5">ie</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></span>
;<br/>
<a NAME="E4:26_1_3"/><span class="lab"><font color="Green" title="E40">A64</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_2.html#T6" target="_self" title="RADIX_2:th.6">Th6</a></span>;<br/>
<span class="lab"><font color="Green" title="E41">A65</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> = 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></span>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_3"><span class="lab"><font color="Green" title="E40">A64</font></span></a>, <a class="ref" href="xcmplx_1.html#T114" title="XCMPLX_1:th.114">XCMPLX_1:114</a></span>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T51" title="INT_1:th.51">INT_1:51</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></span>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:26_1"><span class="lab"><font color="Green" title="E36">A60</font></span></a>, <a class="txt" href="#E3:26_1_3"><span class="lab"><font color="Green" title="E39">A63</font></span></a>, <a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a>, <a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></span>
;<br/>
<a NAME="E6:26_1_3"/><span class="lab"><font color="Green" title="E42">A66</font></span>: 
 <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_2.html#T6" target="_self" title="RADIX_2:th.6">Th6</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c11">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c11">i</font> &amp; <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">en</font> implies <font color="Maroon" title="c9">en</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E7:26_1_3"/><span class="lab"><font color="Green" title="E43">A67</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font>
 <span class="kw">and </span><br/><a NAME="E8:26_1_3"/><span class="lab"><font color="Green" title="E44">A68</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">en</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">en</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span><br/>

<a NAME="E9:26_1_3"/><span class="lab"><font color="Green" title="E45">A69</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_1_3"><span class="lab"><font color="Green" title="E43">A67</font></span></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/>
<a NAME="E10:26_1_3"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <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="#E9:26_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E7:26_1_3"><span class="lab"><font color="Green" title="E43">A67</font></span></a>, <a class="txt" href="#E8:26_1_3"><span class="lab"><font color="Green" title="E44">A68</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/>
<a NAME="E11:26_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A70</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E12:26_1_3"/><span class="lab"><font color="Green" title="E47">A71</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E7:26_1_3"><span class="lab"><font color="Green" title="E43">A67</font></span></a>, <a class="txt" href="#E8:26_1_3"><span class="lab"><font color="Green" title="E44">A68</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c9">en</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">i</font> = 
<font color="Maroon" title="c6">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E11:26_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a></span>
<br/>.= 
 <a href="radix_2.html#K4" title="RADIX_2:func.4">DigitDC2</a> <font color="Maroon" title="c5">ie</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E11:26_1_3"><span class="lab"><font color="Green" title="E46">A70</font></span></a>, <a class="ref" href="radix_2.html#D5" target="_self" title="RADIX_2:def.5">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:26_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E9:26_1_3"><span class="lab"><font color="Green" title="E45">A69</font></span></a>, <a class="ref" href="radix_2.html#T4" target="_self" title="RADIX_2:th.4">Th4</a>, <a class="ref" href="radix_2.html#T6" target="_self" title="RADIX_2:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:26_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E2:26_1_3"><span class="lab"><font color="Green" title="E38">A62</font></span></a>, <a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_1_3"><span class="lab"><font color="Green" title="E43">A67</font></span></a>, <a class="txt" href="#E6:26_1_3"><span class="lab"><font color="Green" title="E42">A66</font></span></a>, <a class="ref" href="pepin.html#T27" title="PEPIN:th.27">PEPIN:27</a></span>
<br/>.= 
<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" title="c6">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T29" title="NAT_2:th.29">NAT_2:29</a></span>
<br/>.= 
 <a href="radix_2.html#K4" title="RADIX_2:func.4">DigitDC2</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_1_3"><span class="lab"><font color="Green" title="E43">A67</font></span></a>, <a class="txt" href="#E5:26_1_3"><span class="lab"><font color="Green" title="E41">A65</font></span></a>, <a class="ref" href="radix_2.html#T4" target="_self" title="RADIX_2:th.4">Th4</a>, <a class="ref" href="radix_2.html#T6" target="_self" title="RADIX_2:th.6">Th6</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E14:26_1_3"/>
<font color="Maroon" title="c9">en</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26_1_3"><span class="lab"><font color="Green" title="E47">A71</font></span></a>, <a class="ref" href="radix_2.html#D5" target="_self" title="RADIX_2:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E27:26_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">en</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c3">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E28:26_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A72</font></span>: 
<font color="Maroon" title="c9">en</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:26_1"><span class="lab"><font color="Green" title="E37">A61</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/>
<a NAME="E29:26_1"/>
( <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E30:26_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">I1</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="radix_2.html#D9" target="_self" title="RADIX_2:def.9">Def9</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E24:26_1"><span class="lab"><font color="Green" title="E35">A41</font></span></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>
<br/>.= 
<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" title="c2">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p0">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E5:26_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E22:26_1"><span class="lab"><font color="Green" title="E33">A39</font></span></a>, <a class="txt" href="#E28:26_1"><span class="lab"><font color="Green" title="E38">A72</font></span></a></span>
<br/>.= 
<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" title="c2">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p5">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T12" title="PEPIN:th.12">PEPIN:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#T10" title="EULER_2:th.10">EULER_2:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T14" title="NEWTON:th.14">NEWTON:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#T9" title="EULER_2:th.9">EULER_2:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c9">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E32:26_1"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">ie</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:26_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:26"/><span class="lab"><font color="Green" title="E19">A73</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">k</font>, <font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">ie</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">ie</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> 1,<font color="Maroon" title="c2">k</font> &amp; <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies for <font color="Olive" title="b1">e</font> being  <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of 1, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <font color="Maroon" title="c4">ie</font>,1,<font color="Maroon" title="c2">k</font> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c1">m</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">ie</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">f</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:26_2"/><span class="lab"><font color="Green" title="E20">A74</font></span>: 
<font color="Maroon" title="c4">ie</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> 1,<font color="Maroon" title="c2">k</font>
 <span class="kw">and </span><br/><a NAME="E2:26_2"/>
<font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">e</font> being  <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of 1, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <font color="Maroon" title="c4">ie</font>,1,<font color="Maroon" title="c2">k</font> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c1">m</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">ie</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">f</font></span><br/>

<a NAME="E3:26_2"/>
<font color="Maroon" title="c4">ie</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_2"><span class="lab"><font color="Green" title="E20">A74</font></span></a>, <a class="ref" href="radix_1.html#D12" title="RADIX_1:def.12">RADIX_1:def 12</a></span>;<br/>
<a NAME="E4:26_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A75</font></span>: 
<font color="Maroon" title="c4">ie</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">e</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of 1, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <font color="Maroon" title="c4">ie</font>,1,<font color="Maroon" title="c2">k</font> implies <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">ie</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">f</font> )</span><br/>

<a NAME="E5:26_2"/><span class="lab"><font color="Green" title="E22">A76</font></span>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E6:26_2"/><span class="lab"><font color="Green" title="E23">A77</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E7:26_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <font color="Maroon" title="c4">ie</font>,1,<font color="Maroon" title="c2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E8:26_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A78</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <font color="Maroon" title="c4">ie</font>,1,<font color="Maroon" title="c2">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_2"><span class="lab"><font color="Green" title="E23">A77</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E9:26_2"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <font color="Maroon" title="c4">ie</font>,1,<font color="Maroon" title="c2">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">ie</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">f</font></span><br/>

<span class="kw">then </span><font color="Maroon" title="c5">e</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <font color="Maroon" title="c4">ie</font>,1,<font color="Maroon" title="c2">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26_2"><span class="lab"><font color="Green" title="E24">A78</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
 <a href="radix_2.html#K4" title="RADIX_2:func.4">DigitDC2</a> <font color="Maroon" title="c4">ie</font>,1,<font color="Maroon" title="c2">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_2"><span class="lab"><font color="Green" title="E23">A77</font></span></a>, <a class="ref" href="radix_2.html#D5" target="_self" title="RADIX_2:def.5">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">ie</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_2"><span class="lab"><font color="Green" title="E22">A76</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">ie</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></span>
<br/>.= 
<font color="Maroon" title="c4">ie</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T6" title="NAT_2:th.6">NAT_2:6</a></span>
<br/>.= 
<font color="Maroon" title="c4">ie</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></span>
<br/>.= 
<font color="Maroon" title="c4">ie</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_2"><span class="lab"><font color="Green" title="E21">A75</font></span></a>, <a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></span>
;<br/>
<a NAME="E11:26_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">ie</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c3">f</font>,1
 
;<br/>
<span class="kw">hence </span><a NAME="E12:26_2"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">ie</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="radix_2.html#D9" target="_self" title="RADIX_2:def.9">Def9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:26"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S8" title="NAT_1:sch.8">NAT_1:sch 8</a>(<a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E19">A73</font></span></a>, <a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E18">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:26"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/>for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">ie</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">ie</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b3">k</font> &amp; <font color="Olive" title="b4">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>for <font color="Olive" title="b6">e</font> being  <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Olive" title="b1">n</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <font color="Olive" title="b5">ie</font>,<font color="Olive" title="b1">n</font>,<font color="Olive" title="b3">k</font> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b6">e</font>,<font color="Olive" title="b4">f</font>,<font color="Olive" title="b3">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b5">ie</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Olive" title="b4">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
