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

<b>let </b><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 implies for <font color="Olive" title="b1">r</font> being  <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> </span>)</span> holds   <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" title="RADIX_6:func.4">Mmax</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Olive" title="b1">r</font> )</span><br/>

<b>assume </b><a NAME="E1:10"/><i><font color="Green" title="E10">A1</font></i>: 
( <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> </span>)</span> holds   <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" title="RADIX_6:func.4">Mmax</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Olive" title="b1">r</font></span><br/>

<a NAME="E2:10"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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/>
<b>let </b><font color="Maroon" title="c3">r</font> be   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" title="RADIX_6:func.4">Mmax</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c3">r</font></span><br/>

<a NAME="E3:10"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> holds <br/> <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" title="RADIX_6:func.4">Mmax</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c3">r</font>,<font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">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="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> implies  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" title="RADIX_6:func.4">Mmax</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">i</font> )</span><br/>

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c4">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="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" title="RADIX_6:func.4">Mmax</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">i</font></span><br/>

<a NAME="E2:10_1"/><b>then </b><i><font color="Green" title="E13">A4</font></i>: 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" title="RADIX_6:func.4">Mmax</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_6.html#K3" title="RADIX_6:func.3">MmaxDigit</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">i</font>
 
<b>by </b><i><a class="ref" href="radix_6.html#D4" target="_self" title="RADIX_6:def.4">Def4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1_1"/>
( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> or <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_1"><b>suppose </b></a><a NAME="E1:10_1_1_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" title="RADIX_6:func.4">Mmax</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">i</font></span><br/><div class="add"><b>then </b> <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" title="RADIX_6:func.4">Mmax</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">i</font> = 
<font color="Maroon" title="c3">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="#E1:10_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="#E2:10_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="radix_6.html#D3" target="_self" title="RADIX_6:def.3">Def3</a></i>
<br/>.= 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="txt" href="#E1:10_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="radix_1.html#D3" title="RADIX_1:def.3">RADIX_1:def 3</a></i>
;<br/><b>hence </b><a NAME="E3:10_1_1_1_1"/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" title="RADIX_6:func.4">Mmax</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_2"><b>suppose </b></a><a NAME="E1:10_1_1_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" title="RADIX_6:func.4">Mmax</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">i</font></span><br/><div class="add"><a NAME="E2:10_1_1_1_2"/><b>then </b><i><font color="Green" title="E14">A5</font></i>: 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" title="RADIX_6:func.4">Mmax</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="#E1:10_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="#E2:10_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="radix_6.html#D3" target="_self" title="RADIX_6:def.3">Def3</a></i>;<br/><a NAME="E3:10_1_1_1_2"/><i><font color="Green" title="E15">A6</font></i>: 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="#E1:10_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="radix_1.html#D3" title="RADIX_1:def.3">RADIX_1:def 3</a></i>;<br/><a NAME="E4:10_1_1_1_2"/>
<font color="Maroon" title="c3">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> 
 <b>by </b><i><a class="txt" href="#E1:10_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="radix_1.html#T18" title="RADIX_1:th.18">RADIX_1:18</a></i>;<br/><b>hence </b><a NAME="E5:10_1_1_1_2"/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" title="RADIX_6:func.4">Mmax</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="#E2:10_1_1_1_2"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="#E3:10_1_1_1_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="radix_1.html#T15" title="RADIX_1:th.15">RADIX_1:15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:10_1"/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" title="RADIX_6:func.4">Mmax</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:10"/>
 <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_6.html#K4" title="RADIX_6:func.4">Mmax</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="#E2:10"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="radix_5.html#T13" title="RADIX_5:th.13">RADIX_5:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
