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

<span class="kw">let </span><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies  for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">k</font> holds <br/><font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Olive" title="b1">m</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">k</font>)</span>)</span> )</span><br/>



<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> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> $1,<font color="Maroon" title="c1">k</font> holds <br/><font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Olive" title="b1">m</font>,$1,<font color="Maroon" title="c1">k</font>)</span>)</span>;<br/>
<a NAME="E1:35"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
 for <font color="Olive" title="b1">u</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">u</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">u</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">u</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">u</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="c3">u</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">u</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="E1:35_1"/>
<font color="Maroon" title="c3">u</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">and </span><br/><a NAME="E2:35_1"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">u</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E3:35_1"/>
 for <font color="Olive" title="b1">p</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">p</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c1">k</font> holds <br/><font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Olive" title="b1">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">k</font>)</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">p</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c1">k</font> implies <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c4">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">k</font>)</span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">I</font> = <font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">R</font> = <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">M</font> = <font color="Maroon" title="c4">p</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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">N</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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:35_1_1"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">u</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 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</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/>
<a NAME="E2:35_1_1"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon" title="c3">u</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c4">p</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E3:35_1_1"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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/>
<a NAME="E4:35_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A6</font></span>: 
<font color="Maroon" title="c3">u</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="c3">u</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="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:35_1_1"/>
<font color="Maroon" title="c4">p</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c1">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c4">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">k</font>)</span>)</span></span><br/>

<a NAME="E6:35_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">p</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="c1">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</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="ref" href="radix_1.html#D12" target="_self" title="RADIX_1:def.12">Def12</a></span>;<br/>
<a NAME="E7:35_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">p</font> <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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K9" title="RADIX_1:func.9">DigitDC</a> (<font color="Maroon" title="c4">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E31">A7</font></span>: <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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span> = 
 <a href="radix_1.html#K6" title="RADIX_1:func.6">SubDigit</a> (<span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c4">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">k</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E4:35_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="radix_1.html#D9" target="_self" title="RADIX_1:def.9">Def9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c4">p</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</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="#E4:35_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="radix_1.html#D6" target="_self" title="RADIX_1:def.6">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c4">p</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</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="#E2:35_1_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>
;<br/>
<a NAME="E9:35_1_1"/><span class="lab"><font color="Green" title="E32">A8</font></span>: 
<span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c4">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">k</font>)</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_1_1_2"/>
<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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">z1</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T74" title="FINSEQ_1:th.74">FINSEQ_1:74</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">DD</font> =  <a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c4">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">k</font>)</span>)</span> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">z0</font> =  <a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c1">k</font>)</span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">z</font> = <span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">z1</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/>
<span class="lab"><font color="Green" title="E33">A9</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">z</font> = 
<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_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</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"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>
<br/>.= 
<font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>
;<br/>
<a NAME="E6:35_1_1_2"/><span class="lab"><font color="Green" title="E34">A10</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="c12">z</font> holds <br/><font color="Maroon" title="c12">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">DD</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c13">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="c12">z</font> implies <font color="Maroon" title="c12">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">DD</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_1_2_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c13">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="c12">z</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">DD</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font></span><br/>

<a NAME="E2:35_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A11</font></span>: 
<font color="Maroon" title="c13">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="c3">u</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="#E5:35_1_1_2"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_2_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="E1:35_1_1_2_2_1"/>
( <font color="Maroon" title="c13">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="c3">u</font> or <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_2_2"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="ref" href="finseq_2.html#T7" title="FINSEQ_2:th.7">FINSEQ_2:7</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E36">A12</font></span>: 
<font color="Maroon" title="c13">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="c3">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">DD</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font></span><br/><div class="add"><a NAME="E2:35_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span> <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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">i</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E2:35_1_1_2_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_1.html#T6" target="_self" title="RADIX_1:th.6">Th7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:35_1_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E38">A14</font></span>: 
<span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</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_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/>
<a NAME="E5:35_1_1_2_2_1_1_1"/>
 <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">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="power.html#T34" title="POWER:th.34">POWER:34</a></span>;<br/>
<a NAME="E6:35_1_1_2_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">t</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="txt" href="#E4:35_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="ref" href="prepower.html#T5" title="PREPOWER:th.5">PREPOWER:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:35_1_1_2_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span> <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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="ref" href="radix_1.html#T3" target="_self" title="RADIX_1:th.3">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:35_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E38">A15</font></span>: 
<font color="Maroon" title="c13">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"><span class="p2">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">z1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_2"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E2:35_1_1_2_2"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E4:35_1_1_2_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">DD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><a NAME="E5:35_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A16</font></span>: 
<font color="Maroon" title="c13">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="c10">DD</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_2_2"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E6:35_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A17</font></span>: 
<font color="Maroon" title="c10">DD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">DD</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><span class="lab"><font color="Green" title="E41">A18</font></span>: <font color="Maroon" title="c10">DD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> = 
<span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c4">p</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="radix_1.html#K6" title="RADIX_1:func.6">SubDigit</a> (<span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c4">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">k</font>)</span>)</span>,<font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c1">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_2_2"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="ref" href="radix_1.html#D6" target="_self" title="RADIX_1:def.6">Def6</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="c1">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K9" title="RADIX_1:func.9">DigitDC</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c1">k</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_2_2"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="ref" href="radix_1.html#D9" target="_self" title="RADIX_1:def.9">Def9</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="c1">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">i</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="c1">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>

;<br/><a NAME="E8:35_1_1_2_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><a NAME="E9:35_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A19</font></span>: 
<font color="Maroon" title="c13">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_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E36">A12</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"><span class="p2">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">z1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> = 
<span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="radix_1.html#K6" title="RADIX_1:func.6">SubDigit</a> (<span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c1">k</font>)</span>)</span>,<font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c1">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="ref" href="radix_1.html#D6" target="_self" title="RADIX_1:def.6">Def6</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="c1">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K9" title="RADIX_1:func.9">DigitDC</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c1">k</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="ref" href="radix_1.html#D9" target="_self" title="RADIX_1:def.9">Def9</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="c1">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">i</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="c1">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E11:35_1_1_2_2_1_1"/>
<font color="Maroon" title="c12">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">DD</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E6:35_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="txt" href="#E3:35_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E36">A20</font></span>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">DD</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c12">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font> = 
<font color="Maroon" title="c12">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</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="#E3:35_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E5:35_1_1_2"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">z1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span></span>)</span> <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="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c4">p</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</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="#E8:35_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="finseq_1.html#T42" title="FINSEQ_1:th.42">FINSEQ_1:42</a></span>
<br/>.= 
<font color="Maroon" title="c10">DD</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E1:35_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E36">A20</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E7:35_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">DD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:35_1_1_2"/>
<span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c4">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">k</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_2"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E6:35_1_1_2"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="ref" href="finseq_5.html#T13" title="FINSEQ_5:th.13">FINSEQ_5:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:35_1_1"/>
 <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">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="power.html#T34" title="POWER:th.34">POWER:34</a></span>;<br/>
<a NAME="E11:35_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A21</font></span>: 
<span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</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="prepower.html#T5" title="PREPOWER:th.5">PREPOWER:5</a></span>;<br/>
<a NAME="E12:35_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">p</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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></span>;<br/>
<a NAME="E13:35_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A22</font></span>: 
<font color="Maroon" title="c4">p</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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_1.html#D12" target="_self" title="RADIX_1:def.12">Def12</a></span>;<br/>
<a NAME="E14:35_1_1"/>
<font color="Maroon" title="c4">p</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"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></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="c4">p</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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:35_1_1"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></span>;<br/>
<a NAME="E15:35_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</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="c1">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E13:35_1_1"><span class="lab"><font color="Green" title="E34">A22</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:35_1_1"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c4">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">k</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="rvsum_1.html#T74" title="RVSUM_1:th.74">RVSUM_1:74</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:35_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:35"/><span class="lab"><font color="Green" title="E26">A23</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">m</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">m</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> 1,<font color="Maroon" title="c1">k</font> implies <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c3">m</font>,1,<font color="Maroon" title="c1">k</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_2"/><span class="lab"><font color="Green" title="E27">A24</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> 1,<font color="Maroon" title="c1">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c3">m</font>,1,<font color="Maroon" title="c1">k</font>)</span>)</span></span><br/>

<a NAME="E2:35_2"/><span class="lab"><font color="Green" title="E28">A25</font></span>: 
<font color="Maroon" title="c3">m</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">M</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T74" title="FINSEQ_1:th.74">FINSEQ_1:74</a></span>;<br/>
<a NAME="E4:35_2"/><span class="lab"><font color="Green" title="E29">A26</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#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
 <a href="radix_1.html#K6" title="RADIX_1:func.6">SubDigit</a> (<span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c3">m</font>,1,<font color="Maroon" title="c1">k</font>)</span>)</span>,1,<font color="Maroon" title="c1">k</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="c1">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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K5" title="RADIX_1:func.5">DigB</a> (<span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c3">m</font>,1,<font color="Maroon" title="c1">k</font>)</span>)</span>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T232" title="XREAL_1:th.232">XREAL_1:232</a></span>
<br/>.= 
1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K5" title="RADIX_1:func.5">DigB</a> (<span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c3">m</font>,1,<font color="Maroon" title="c1">k</font>)</span>)</span>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T4" title="NEWTON:th.4">NEWTON:4</a></span>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2"><span class="lab"><font color="Green" title="E27">A24</font></span></a>, <a class="ref" href="radix_1.html#T21" target="_self" title="RADIX_1:th.21">Th24</a></span>
;<br/>
<a NAME="E6:35_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c3">m</font>,1,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_2"><span class="lab"><font color="Green" title="E29">A26</font></span></a>, <a class="ref" href="radix_1.html#D6" target="_self" title="RADIX_1:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span> <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c3">m</font>,1,<font color="Maroon" title="c1">k</font>)</span>)</span> = 
 <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <font color="Maroon" title="c4">M</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_1.html#T17" target="_self" title="RADIX_1:th.17">Th20</a></span>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_2"><span class="lab"><font color="Green" title="E28">A25</font></span></a>, <a class="ref" href="finsop_1.html#T11" title="FINSOP_1:th.11">FINSOP_1:11</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:35"/>
 for <font color="Olive" title="b1">u</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">u</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">u</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="#E2:35"><span class="lab"><font color="Green" title="E26">A23</font></span></a>, <a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E25">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:35"/>
( <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies  for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">k</font> holds <br/><font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Olive" title="b1">m</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">k</font>)</span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
