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

<b>let </b><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">n</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="E1:4"/>
( <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c2">x</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c1">k</font> )
 ;<br/>

<a NAME="E2:4"/><b>then </b>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E3:4"/><b>then </b><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">xn</font> = <font color="Maroon" title="c2">x</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="c3">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">Rn</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#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c3">n</font>;<br/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</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="c2">x</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="c3">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c3">n</font> = 
 <a href="radix_1.html#K9" title="RADIX_1:func.9">DigitDC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</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#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">k</font>
<b>by </b><i><a class="txt" href="radix_4.html#E3:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="radix_1.html#D9" title="RADIX_1:def.9">RADIX_1:def 9</a></i>
<br/>.= 
 <a href="radix_1.html#K9" title="RADIX_1:func.9">DigitDC</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">k</font>
<b>by </b><i><a class="ref" href="euler_2.html#T7" title="EULER_2:th.7">EULER_2:7</a></i>
<br/>.= 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c3">n</font>
<b>by </b><i><a class="txt" href="radix_4.html#E3:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="radix_1.html#D9" title="RADIX_1:def.9">RADIX_1:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E5:4"/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</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="c2">x</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="c3">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c3">n</font>
 ;<br/>


</div>
