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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><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="E1:13"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:13"/><b>then </b><i><font color="Green">E31</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:13"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E4:13"/><b>then </b><i><font color="Green">E34</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:13"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E6:13"/><b>then </b><i><font color="Green">E43</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"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>assume </b><a NAME="E7:13"/><i><font color="Green">E44</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<b>set </b><font color="Maroon">X</font> = <font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">Y</font> = <font color="Maroon">y</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span>;<br/>
<a NAME="E8:13"/><i><font color="Green">E45</font></i>: 
 <a href="radix_4.html#K1" target="_self">SDSubAddDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">y</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_3.html#K6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">y</font>,<span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">y</font>,<span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1"><b>suppose </b></a><a NAME="E1:13_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green">E65</font></i>:  <a href="radix_3.html#K7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
 <a href="radix_3.html#K7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,0
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="radix_3.html#D5">RADIX_3:def 5</a></i>
;<br/><i><font color="Green">E66</font></i>:  <a href="radix_3.html#K7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">y</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
 <a href="radix_3.html#K7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">y</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,0
<b>by </b><i>, <a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="radix_3.html#D5">RADIX_3:def 5</a></i>
;<br/><a NAME="E4:13_1_1_1"/><i><font color="Green">E67</font></i>: 
 <a href="radix_3.html#K7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_3.html#K7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="radix_3.html#T21">RADIX_3:21</a></i>;<br/><a NAME="E5:13_1_1_1"/><i><font color="Green">E68</font></i>: 
 <a href="radix_3.html#K7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">y</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_3.html#K7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="radix_3.html#T21">RADIX_3:21</a></i>;<br/><i><font color="Green">E69</font></i>:  <a href="radix_3.html#K7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
 <a href="radix_3.html#K7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,0
<b>by </b><i>, <a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="radix_3.html#D5">RADIX_3:def 5</a></i>
;<br/> <a href="radix_3.html#K7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
 <a href="radix_3.html#K7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,0
<b>by </b><i>, <a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="radix_3.html#D5">RADIX_3:def 5</a></i>
;<br/><b>hence </b><a NAME="E8:13_1_1_1"/>
 <a href="radix_4.html#K1" target="_self">SDSubAddDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">y</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="radix_4.html#K1" target="_self">SDSubAddDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">k</font>
 <b>by </b><i>, , , , , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_2"><b>suppose </b></a><a NAME="E1:13_1_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:13_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:13_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:13_1_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan12.html#T1">JORDAN12:1</a></i>;<br/><a NAME="E5:13_1_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E6:13_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:13_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E8:13_1_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/> <a href="radix_4.html#K1" target="_self">SDSubAddDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">y</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><a href="radix_3.html#K6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">y</font>,<span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">y</font>,<span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="radix_3.html#T21">RADIX_3:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">y</font>,<span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="radix_3.html#T21">RADIX_3:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">y</font>,<span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="radix_3.html#T21">RADIX_3:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="radix_3.html#T21">RADIX_3:21</a></i>
<br/>.= 
 <a href="radix_4.html#K1" target="_self">SDSubAddDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">k</font>
<b>by </b><i>, , </i>
;<br/><b>hence </b><a NAME="E10:13_1_1_2"/>
 <a href="radix_4.html#K1" target="_self">SDSubAddDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">y</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="radix_4.html#K1" target="_self">SDSubAddDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:13"/>
 <a href="radix_4.html#K1" target="_self">SDSubAddDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">y</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="radix_4.html#K1" target="_self">SDSubAddDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">k</font>
 ;<br/>


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