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

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:17"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 ;<br/>

<a NAME="E2:17"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<b>let </b><font color="Maroon">r</font> be   <a href="binarith.html#NM1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a> </span>)</span>;<br/>

<a NAME="E3:17"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <br/>(  not <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> or (  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K2" target="_self">M0</a> <font color="Maroon">r</font></span>)</span>,<font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" target="_self">Mmax</a> <font color="Maroon">r</font></span>)</span>,<font color="Olive">i</font> &amp;  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K4">SDMax</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Olive">i</font> <a href="hidden.html#R1">=</a> 0 ) or (  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K4">SDMax</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" target="_self">Mmax</a> <font color="Maroon">r</font></span>)</span>,<font color="Olive">i</font> &amp;  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K2" target="_self">M0</a> <font color="Maroon">r</font></span>)</span>,<font color="Olive">i</font> <a href="hidden.html#R1">=</a> 0 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 ;<br/>

<a NAME="E2:17_1"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:17_1"/><i><font color="Green">E29</font></i>: 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" target="_self">Mmax</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_6.html#K3" target="_self">MmaxDigit</a> <font color="Maroon">r</font>,<font color="Maroon">i</font>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green">E31</font></i>:  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" target="_self">Mmax</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i>, , <a class="ref" href="radix_6.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
 <a href="radix_5.html#K3">SDMaxDigit</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>,<font color="Maroon">i</font>
<b>by </b><i>, <a class="txt" href="radix_6.html#E1:17"><i><font color="Green">E20</font></i></a>, , <a class="ref" href="radix_5.html#D3">RADIX_5:def 3</a></i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K4">SDMax</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="radix_5.html#D4">RADIX_5:def 4</a></i>
;<br/> <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K2" target="_self">M0</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">i</font> = 
 <a href="radix_6.html#K1" target="_self">M0Digit</a> <font color="Maroon">r</font>,<font color="Maroon">i</font>
<b>by </b><i>, </i>
<br/>.= 
0
<b>by </b><i>, , </i>
;<br/><b>hence </b><a NAME="E4:17_1_1_1_1"/>
( (  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K2" target="_self">M0</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" target="_self">Mmax</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">i</font> &amp;  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K4">SDMax</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 ) or (  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K4">SDMax</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" target="_self">Mmax</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">i</font> &amp;  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K2" target="_self">M0</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green">E33</font></i>:  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" target="_self">Mmax</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">i</font> = 
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="radix_6.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
 <a href="radix_6.html#K1" target="_self">M0Digit</a> <font color="Maroon">r</font>,<font color="Maroon">i</font>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K2" target="_self">M0</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">i</font>
<b>by </b><i>, </i>
;<br/> <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K4">SDMax</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> = 
 <a href="radix_5.html#K3">SDMaxDigit</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>,<font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="radix_5.html#D4">RADIX_5:def 4</a></i>
<br/>.= 
0
<b>by </b><i>, , <a class="ref" href="radix_5.html#D3">RADIX_5:def 3</a></i>
;<br/><b>hence </b><a NAME="E4:17_1_1_1_2"/>
( (  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K2" target="_self">M0</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" target="_self">Mmax</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">i</font> &amp;  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K4">SDMax</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 ) or (  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K4">SDMax</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" target="_self">Mmax</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">i</font> &amp;  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K2" target="_self">M0</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:17_1"/>
( (  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K2" target="_self">M0</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" target="_self">Mmax</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">i</font> &amp;  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K4">SDMax</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 ) or (  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K4">SDMax</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" target="_self">Mmax</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">i</font> &amp;  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K2" target="_self">M0</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 ) )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:17"/>
<span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <span class="p2">(<span class="default"><a href="radix_6.html#K4" target="_self">Mmax</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> 0,<span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <span class="p2">(<span class="default"><a href="radix_6.html#K2" target="_self">M0</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <span class="p2">(<span class="default"><a href="radix_5.html#K4">SDMax</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="radix_6.html#E1"><i><font color="Green">Lemma18</font></i></a>, <a class="ref" href="radix_5.html#T15">RADIX_5:15</a></i>;<br/>


</div>
