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

<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">k</font>, <font color="Maroon" title="c3">m</font>, <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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> <font color="Maroon" title="c1">n</font> &amp; 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font> implies  <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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="c3">m</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="c2">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:40"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<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> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font> or  <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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="c3">m</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="c2">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">i</font> = <font color="Maroon" title="c4">i</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E3:40"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E4:40"/>
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E5:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E6:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<font color="Maroon" title="c5">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">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E7:40"/>
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E8:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A5</font></span>: 
<font color="Maroon" title="c5">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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">M</font> = <font color="Maroon" title="c3">m</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="c2">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E9:40"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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="c3">m</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="c2">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font></span><br/>

<span class="lab"><font color="Green" title="E35">A7</font></span>:  <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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="c3">m</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="c2">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">i</font> = 
 <a href="radix_3.html#K9" title="RADIX_3:func.9">SD2SDSubDigitS</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="c3">m</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="c2">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="radix_3.html#D8" target="_self" title="RADIX_3:def.8">Def8</a></span>
<br/>.= 
 <a href="radix_3.html#K8" title="RADIX_3:func.8">SD2SDSubDigit</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="c3">m</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="c2">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="radix_3.html#D7" target="_self" title="RADIX_3:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</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="c3">m</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="c2">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</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="c3">m</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="c2">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="radix_3.html#D6" target="_self" title="RADIX_3:def.6">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</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="c3">m</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="c2">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="radix_3.html#E5"><span class="lab"><font color="Green" title="E5">Lm5</font></span></a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_2_1"/>
( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_2_1_1"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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="c3">m</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="c2">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font></span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A9</font></span>:  <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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="c3">m</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="c2">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">i</font> = 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</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="c3">m</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="c2">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c2">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_1.html#D3" title="RADIX_1:def.3">RADIX_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_3.html#T17" target="_self" title="RADIX_3:th.17">Th17</a></span>
;<br/> <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">i</font> = 
 <a href="radix_3.html#K9" title="RADIX_3:func.9">SD2SDSubDigitS</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="radix_3.html#D8" target="_self" title="RADIX_3:def.8">Def8</a></span>
<br/>.= 
 <a href="radix_3.html#K8" title="RADIX_3:func.8">SD2SDSubDigit</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="radix_3.html#D7" target="_self" title="RADIX_3:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="radix_3.html#D6" target="_self" title="RADIX_3:def.6">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_2_1_1"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c2">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_1.html#D3" title="RADIX_1:def.3">RADIX_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_3.html#T17" target="_self" title="RADIX_3:th.17">Th17</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:40_2_1_1"/>
 <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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="c3">m</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="c2">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_2_1_1"><span class="lab"><font color="Green" title="E37">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_2_1_2"/><span class="lab"><font color="Green" title="E36">A10</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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="c3">m</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="c2">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font></span><br/><div class="add"><a NAME="E2:40_2_1_2"/>
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E3:40_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A11</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>;<br/><a NAME="E4:40_2_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E1:40_2_1_2"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:40_2_1_2"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a>, <a class="ref" href="jordan12.html#T1" title="JORDAN12:th.1">JORDAN12:1</a></span>;<br/><a NAME="E6:40_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_2_1_2"><span class="lab"><font color="Green" title="E37">A11</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><span class="kw">then </span> <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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="c3">m</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="c2">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">i</font> = 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="txt" href="radix_3.html#E5"><span class="lab"><font color="Green" title="E5">Lm5</font></span></a></span>
<br/>.= 
 <a href="radix_3.html#K8" title="RADIX_3:func.8">SD2SDSubDigit</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="radix_3.html#D6" target="_self" title="RADIX_3:def.6">Def6</a></span>
<br/>.= 
 <a href="radix_3.html#K9" title="RADIX_3:func.9">SD2SDSubDigitS</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c2">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="radix_3.html#D7" target="_self" title="RADIX_3:def.7">Def7</a></span>
<br/>.= 
 <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="radix_3.html#D8" target="_self" title="RADIX_3:def.8">Def8</a></span>
;<br/><span class="kw">hence </span><a NAME="E8:40_2_1_2"/>
 <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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="c3">m</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="c2">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:40"/>
 <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</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="c3">m</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="c2">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
