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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>







<b>assume </b><a NAME="E1:39"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:39"/><i><font color="Green">E30</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K4">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">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E3:39"/><i><font color="Green">E31</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="radix_3.html#E1:39"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:39"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></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="E5:39"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="radix_3.html#E3:39"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:39"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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="txt" href="radix_3.html#E4:39"><i><font color="Green">E32</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E7:39"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></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="txt" href="radix_3.html#E3:39"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<i><font color="Green">E35</font></i>:  <a href="radix_3.html#K7" target="_self">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" target="_self">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">c<sub>3</sub></font> <a href="nat_1.html#K4">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">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font> = 
 <a href="radix_3.html#K9" target="_self">SD2SDSubDigitS</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K4">mod</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="radix_3.html#E5:39"><i><font color="Green">E33</font></i></a>, <a class="ref" href="radix_3.html#D8" target="_self">Def8</a></i>
<br/>.= 
 <a href="radix_3.html#K8" target="_self">SD2SDSubDigit</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K4">mod</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="radix_3.html#E2:39"><i><font color="Green">E30</font></i></a>, <a class="txt" href="radix_3.html#E5:39"><i><font color="Green">E33</font></i></a>, <a class="ref" href="radix_3.html#D7" target="_self">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" target="_self">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E1:39"><i><font color="Green">E29</font></i></a>, <a class="ref" href="radix_3.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" target="_self">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E1:39"><i><font color="Green">E29</font></i></a>, <a class="txt" href="radix_3.html#E5"><i><font color="Green">Lemma5</font></i></a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>4</sub></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="39_2_1_1"><b>suppose </b></a><a NAME="E1:39_2_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green">E37</font></i>:  <a href="radix_3.html#K7" target="_self">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" target="_self">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">c<sub>3</sub></font> <a href="nat_1.html#K4">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">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" target="_self">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,0</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E8:39"><i><font color="Green">E35</font></i></a>, <a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" target="_self">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> 0,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" target="_self">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
<b>by </b><i><a class="txt" href="radix_3.html#E2:39"><i><font color="Green">E30</font></i></a>, <a class="ref" href="radix_3.html#T17" target="_self">Th17</a></i>
;<br/> <a href="radix_3.html#K7" target="_self">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font> = 
 <a href="radix_3.html#K9" target="_self">SD2SDSubDigitS</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="radix_3.html#E7:39"><i><font color="Green">E34</font></i></a>, <a class="ref" href="radix_3.html#D8" target="_self">Def8</a></i>
<br/>.= 
 <a href="radix_3.html#K8" target="_self">SD2SDSubDigit</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="radix_3.html#E2:39"><i><font color="Green">E30</font></i></a>, <a class="txt" href="radix_3.html#E7:39"><i><font color="Green">E34</font></i></a>, <a class="ref" href="radix_3.html#D7" target="_self">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" target="_self">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E5:39"><i><font color="Green">E33</font></i></a>, <a class="ref" href="radix_3.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" target="_self">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,0</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E1:39_2_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" target="_self">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> 0,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" target="_self">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
<b>by </b><i><a class="txt" href="radix_3.html#E2:39"><i><font color="Green">E30</font></i></a>, <a class="ref" href="radix_3.html#T17" target="_self">Th17</a></i>
;<br/><b>hence </b><a NAME="E4:39_2_1_1"/>
 <a href="radix_3.html#K7" target="_self">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_3.html#K7" target="_self">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" target="_self">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">c<sub>3</sub></font> <a href="nat_1.html#K4">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">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="radix_3.html#E2:39_2_1_1"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_2"><b>suppose </b></a><a NAME="E1:39_2_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:39_2_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="radix_3.html#E3:39"><i><font color="Green">E31</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:39_2_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan12.html#T1">JORDAN12:1</a></i>;<br/><a NAME="E4:39_2_1_2"/><b>then </b><i><font color="Green">E36</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E5:39_2_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="radix_3.html#E1:39"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:39_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E7:39_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="radix_3.html#E4:39_2_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>then </b> <a href="radix_3.html#K7" target="_self">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" target="_self">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">c<sub>3</sub></font> <a href="nat_1.html#K4">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">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" target="_self">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E8:39"><i><font color="Green">E35</font></i></a>, <a class="txt" href="radix_3.html#E5"><i><font color="Green">Lemma5</font></i></a></i>
<br/>.= 
 <a href="radix_3.html#K8" target="_self">SD2SDSubDigit</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="radix_3.html#E5:39"><i><font color="Green">E33</font></i></a>, <a class="ref" href="radix_3.html#D6" target="_self">Def6</a></i>
<br/>.= 
 <a href="radix_3.html#K9" target="_self">SD2SDSubDigitS</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="radix_3.html#E2:39"><i><font color="Green">E30</font></i></a>, <a class="txt" href="radix_3.html#E7:39"><i><font color="Green">E34</font></i></a>, <a class="ref" href="radix_3.html#D7" target="_self">Def7</a></i>
<br/>.= 
 <a href="radix_3.html#K7" target="_self">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="radix_3.html#E7:39"><i><font color="Green">E34</font></i></a>, <a class="ref" href="radix_3.html#D8" target="_self">Def8</a></i>
;<br/><b>hence </b><a NAME="E9:39_2_1_2"/>
 <a href="radix_3.html#K7" target="_self">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_3.html#K7" target="_self">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" target="_self">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">c<sub>3</sub></font> <a href="nat_1.html#K4">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">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:39"/>
 <a href="radix_3.html#K7" target="_self">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_3.html#K7" target="_self">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" target="_self">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">c<sub>3</sub></font> <a href="nat_1.html#K4">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">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div>
