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

<b>let </b><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>;<br/>



<a NAME="E1:35"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>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="E2:35"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:35_1"/><i><font color="Green" title="E27">A3</font></i>: 
<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>
 ;<br/>

<a NAME="E2:35_1"/><i><font color="Green" title="E28">A4</font></i>: 
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
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<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#K16" title="NEWTON:func.16">|^</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>
<b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>
<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>
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="txt" href="radix_1.html#E1:35_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="radix_1.html#T24" target="_self" title="RADIX_1:th.24">Th24</a></i>
;<br/>
<a NAME="E4:35_1"/><b>then </b><i><font color="Green" title="E29">A5</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="radix_1.html#E2:35_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="radix_1.html#D6" target="_self" title="RADIX_1:def.6">Def6</a></i>;<br/>
<a NAME="E5:35_1"/><i><font color="Green" title="E30">A6</font></i>: 
<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> 
 
<b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<b>then reconsider </b><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>  <b>by </b><i><a class="ref" href="finseq_1.html#T95" title="FINSEQ_1:th.95">FINSEQ_1:95</a></i>;<br/>
<b>thus </b> <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>
<b>by </b><i><a class="txt" href="radix_1.html#E4:35_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="radix_1.html#T20" target="_self" title="RADIX_1:th.20">Th20</a></i>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="txt" href="radix_1.html#E5:35_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:35"/><i><font color="Green" title="E27">A7</font></i>: 
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_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">u</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:35_2"/><i><font color="Green" title="E28">A8</font></i>: 
( <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>] )
 ;<br/>

<a NAME="E2:35_2"/>
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_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">p</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:35_2_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>
 ;<br/>

<a NAME="E2:35_2_1"/><b>then </b><i><font color="Green" title="E29">A9</font></i>: 
<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#K16" title="NEWTON:func.16">|^</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>
 
<b>by </b><i><a class="ref" href="radix_1.html#D12" target="_self" title="RADIX_1:def.12">Def12</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">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/>
<a NAME="E3:35_2_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> 
 
<b>by </b><i><a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>;<br/>
<a NAME="E4:35_2_1"/><b>then </b>
<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> 
 
<b>by </b><i><a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></i>;<br/>
<a NAME="E5:35_2_1"/><b>then </b><i><font color="Green" title="E30">A10</font></i>: 
<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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E6:35_2_1"/><b>then </b><i><font color="Green" title="E31">A11</font></i>: 
<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>
 
<b>by </b><i><a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">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/>
<b>set </b><font color="Maroon" title="c7">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="E7:35_2_1"/>
<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>
 
<b>by </b><i><a class="txt" href="radix_1.html#E5:35_2_1"><i><font color="Green" title="E30">A10</font></i></a>, <a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></i>;<br/>
<a NAME="E8:35_2_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="radix_1.html#D12" target="_self" title="RADIX_1:def.12">Def12</a></i>;<br/>
<a NAME="E9:35_2_1"/><b>then </b><i><font color="Green" title="E32">A12</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="radix_1.html#E1:35_2"><i><font color="Green" title="E28">A8</font></i></a>, <a class="txt" href="radix_1.html#E6:35_2_1"><i><font color="Green" title="E31">A11</font></i></a></i>;<br/>
<a NAME="E10:35_2_1"/>
<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> 
 
<b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">NN</font> = <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&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="binarith.html#K11" title="BINARITH:func.11">*&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>  <b>by </b><i><a class="ref" href="finseq_1.html#T95" title="FINSEQ_1:th.95">FINSEQ_1:95</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">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> <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> 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/>
<b>set </b><font color="Maroon" title="c10">I</font> = <font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<a NAME="E13:35_2_1"/><i><font color="Green" title="E33">A13</font></i>: 
<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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/>
<a NAME="E14:35_2_1"/><b>then </b><i><font color="Green" title="E34">A14</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="radix_1.html#E2:35_2_1"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></i>;<br/>
<a NAME="E15:35_2_1"/><i><font color="Green" title="E35">A15</font></i>: 
( 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 &amp; <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> <font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E16:35_2_1"/><b>then </b><i><font color="Green" title="E36">A16</font></i>: 
<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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E17:35_2_1"/><i><font color="Green" title="E37">A17</font></i>: 
<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>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<i><font color="Green" title="E38">A18</font></i>: <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>
<b>by </b><i><a class="txt" href="radix_1.html#E13:35_2_1"><i><font color="Green" title="E33">A13</font></i></a>, <a class="txt" href="radix_1.html#E14:35_2_1"><i><font color="Green" title="E34">A14</font></i></a>, <a class="txt" href="radix_1.html#E16:35_2_1"><i><font color="Green" title="E36">A16</font></i></a>, <a class="ref" href="radix_1.html#D9" target="_self" title="RADIX_1:def.9">Def9</a></i>
<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>
<b>by </b><i><a class="txt" href="radix_1.html#E16:35_2_1"><i><font color="Green" title="E36">A16</font></i></a>, <a class="ref" href="radix_1.html#D6" target="_self" title="RADIX_1:def.6">Def6</a></i>
<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>
<b>by </b><i><a class="txt" href="radix_1.html#E15:35_2_1"><i><font color="Green" title="E35">A15</font></i></a>, <a class="txt" href="radix_1.html#E17:35_2_1"><i><font color="Green" title="E37">A17</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
;<br/>
<a NAME="E19:35_2_1"/>
<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="binarith.html#K11" title="BINARITH:func.11">&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="binarith.html#K11" title="BINARITH:func.11">*&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_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:35_2_1_2"/><i><font color="Green" title="E39">A19</font></i>: 
<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> 
 
<b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<b>set </b><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/>
<b>reconsider </b><font color="Maroon" title="c12">z1</font> = <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&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="binarith.html#K11" title="BINARITH:func.11">*&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>  <b>by </b><i><a class="txt" href="radix_1.html#E1:35_2_1_2"><i><font color="Green" title="E39">A19</font></i></a>, <a class="ref" href="finseq_1.html#T95" title="FINSEQ_1:th.95">FINSEQ_1:95</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">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/>
<b>reconsider </b><font color="Maroon" title="c14">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="c12">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/>
<i><font color="Green" title="E40">A20</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">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="binarith.html#K11" title="BINARITH:func.11">&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="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<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="binarith.html#K11" title="BINARITH:func.11">&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="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>
<br/>.= 
<font color="Maroon" title="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>
;<br/>
<a NAME="E6:35_2_1_2"/><i><font color="Green" title="E41">A21</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">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
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E7:35_2_1_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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="c14">z</font> holds <br/><font color="Maroon" title="c14">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="c13">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_2_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:35_2_1_2_2"/>
( 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> <font color="Maroon" title="c14">z</font> )
 ;<br/>

<a NAME="E2:35_2_1_2_2"/><b>then </b><i><font color="Green" title="E42">A22</font></i>: 
<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="c3">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="radix_1.html#E5:35_2_1_2"><i><font color="Green" title="E40">A20</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_2_1_2_2_1"/>
( <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> <font color="Maroon" title="c3">u</font> or <font color="Maroon" title="c15">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><b>by </b><i><a class="txt" href="radix_1.html#E2:35_2_1_2_2"><i><font color="Green" title="E42">A22</font></i></a>, <a class="ref" href="finseq_2.html#T8" title="FINSEQ_2:th.8">FINSEQ_2:8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:35_2_1_2_2_1_1"/><i><font color="Green" title="E43">A23</font></i>: 
<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> <font color="Maroon" title="c3">u</font>
 ;<br/><div class="add"><a NAME="E2:35_2_1_2_2_1_1"/><i><font color="Green" title="E44">A24</font></i>: 
<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#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c15">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#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c15">i</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_2_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:35_2_1_2_2_1_1_1"/>
<font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">u</font>
 
<b>by </b><i><a class="txt" href="radix_1.html#E1:35_2_1_2_2_1_1"><i><font color="Green" title="E43">A23</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E2:35_2_1_2_2_1_1_1"/><b>then </b>
<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#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c15">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>
 
<b>by </b><i><a class="ref" href="radix_1.html#T7" target="_self" title="RADIX_1:th.7">Th7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:35_2_1_2_2_1_1_1"/><i><font color="Green" title="E45">A25</font></i>: 
<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#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c15">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c16">t</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<a NAME="E5:35_2_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> 
 
<b>by </b><i><a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>;<br/>
<a NAME="E6:35_2_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c16">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> 
 
<b>by </b><i><a class="txt" href="radix_1.html#E4:35_2_1_2_2_1_1_1"><i><font color="Green" title="E45">A25</font></i></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></i>;<br/>
<b>hence </b><a NAME="E7:35_2_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#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c15">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#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c15">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="radix_1.html#E4:35_2_1_2_2_1_1_1"><i><font color="Green" title="E45">A25</font></i></a>, <a class="ref" href="radix_1.html#T4" target="_self" title="RADIX_1:th.4">Th4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:35_2_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>
 <b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/><a NAME="E4:35_2_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E45">A26</font></i>: 
<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_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>
 <b>by </b><i><a class="txt" href="radix_1.html#E1:35_2_1_2_2_1_1"><i><font color="Green" title="E43">A23</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E46">A27</font></i>: <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="c12">z1</font></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"><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="c15">i</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<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="c15">i</font>
<b>by </b><i><a class="txt" href="radix_1.html#E4:35_2_1_2_2_1_1"><i><font color="Green" title="E45">A26</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<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="c15">i</font>,<font color="Maroon" title="c1">k</font>
<b>by </b><i><a class="txt" href="radix_1.html#E1:35_2_1_2_2_1_1"><i><font color="Green" title="E43">A23</font></i></a>, <a class="ref" href="radix_1.html#D6" target="_self" title="RADIX_1:def.6">Def6</a></i>
<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#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c15">i</font>,<font color="Maroon" title="c1">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_1.html#E1:35_2_1_2_2_1_1"><i><font color="Green" title="E43">A23</font></i></a>, <a class="ref" href="radix_1.html#D9" target="_self" title="RADIX_1:def.9">Def9</a></i>
<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#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c15">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#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="radix_1.html#E2:35_2_1_2_2_1_1"><i><font color="Green" title="E44">A24</font></i></a></i>
;<br/><a NAME="E6:35_2_1_2_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">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
 <b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/><a NAME="E7:35_2_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E47">A28</font></i>: 
<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> <font color="Maroon" title="c13">DD</font>
 <b>by </b><i><a class="txt" href="radix_1.html#E2:35_2_1_2_2"><i><font color="Green" title="E42">A22</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E48">A29</font></i>: <font color="Maroon" title="c13">DD</font> <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"><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="c15">i</font>
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<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="c15">i</font>,<font color="Maroon" title="c1">k</font>
<b>by </b><i><a class="txt" href="radix_1.html#E2:35_2_1_2_2"><i><font color="Green" title="E42">A22</font></i></a>, <a class="ref" href="radix_1.html#D6" target="_self" title="RADIX_1:def.6">Def6</a></i>
<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#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c15">i</font>,<font color="Maroon" title="c1">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_1.html#E2:35_2_1_2_2"><i><font color="Green" title="E42">A22</font></i></a>, <a class="ref" href="radix_1.html#D9" target="_self" title="RADIX_1:def.9">Def9</a></i>
<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#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c15">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#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>

;<br/><a NAME="E9:35_2_1_2_2_1_1"/><i><font color="Green" title="E49">A30</font></i>: 
<font color="Maroon" title="c13">DD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">DD</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="txt" href="radix_1.html#E7:35_2_1_2_2_1_1"><i><font color="Green" title="E47">A28</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E10:35_2_1_2_2_1_1"/>
<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"><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="c12">z1</font></span>)</span>
 <b>by </b><i><a class="txt" href="radix_1.html#E5:35_2_1_2"><i><font color="Green" title="E40">A20</font></i></a>, <a class="txt" href="radix_1.html#E2:35_2_1_2_2"><i><font color="Green" title="E42">A22</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E11:35_2_1_2_2_1_1"/>
<font color="Maroon" title="c14">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">DD</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="txt" href="radix_1.html#E5:35_2_1_2_2_1_1"><i><font color="Green" title="E46">A27</font></i></a>, <a class="txt" href="radix_1.html#E8:35_2_1_2_2_1_1"><i><font color="Green" title="E48">A29</font></i></a>, <a class="txt" href="radix_1.html#E9:35_2_1_2_2_1_1"><i><font color="Green" title="E49">A30</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:35_2_1_2_2_1_2"/><i><font color="Green" title="E43">A31</font></i>: 
<font color="Maroon" title="c15">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
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c14">z</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">i</font> = 
<font color="Maroon" title="c14">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>
<b>by </b><i><a class="txt" href="radix_1.html#E15:35_2_1"><i><font color="Green" title="E35">A15</font></i></a>, <a class="txt" href="radix_1.html#E5:35_2_1_2"><i><font color="Green" title="E40">A20</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<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="c12">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>
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>
<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>
<b>by </b><i><a class="txt" href="radix_1.html#E18:35_2_1"><i><font color="Green" title="E38">A18</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>
<br/>.= 
<font color="Maroon" title="c13">DD</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">i</font>
<b>by </b><i><a class="txt" href="radix_1.html#E15:35_2_1"><i><font color="Green" title="E35">A15</font></i></a>, <a class="txt" href="radix_1.html#E17:35_2_1"><i><font color="Green" title="E37">A17</font></i></a>, <a class="txt" href="radix_1.html#E1:35_2_1_2_2_1_2"><i><font color="Green" title="E43">A31</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E8:35_2_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="binarith.html#K11" title="BINARITH:func.11">&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="binarith.html#K11" title="BINARITH:func.11">*&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>
 <b>by </b><i><a class="txt" href="radix_1.html#E5:35_2_1_2"><i><font color="Green" title="E40">A20</font></i></a>, <a class="txt" href="radix_1.html#E6:35_2_1_2"><i><font color="Green" title="E41">A21</font></i></a>, <a class="ref" href="finseq_5.html#T14" title="FINSEQ_5:th.14">FINSEQ_5:14</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:35_2_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>
 <b>by </b><i><a class="txt" href="radix_1.html#E9:35_2_1"><i><font color="Green" title="E32">A12</font></i></a>, <a class="ref" href="rvsum_1.html#T104" title="RVSUM_1:th.104">RVSUM_1:104</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:35_2"/>
<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]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:35"/>
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>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S8" title="NAT_1:sch.8">NAT_1:sch 8</a>(<a class="txt" href="radix_1.html#E2:35"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="radix_1.html#E3:35"><i><font color="Green" title="E27">A7</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:35"/>
( <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> )
 <b>by </b><i><a class="txt" href="radix_1.html#E1:35"><i><font color="Green" title="E25">A1</font></i></a></i>;<br/>


</div>
