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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">k</font>, <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Olive">x</font> <a href="radix_1.html#R1">is_represented_by</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Olive">k</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="radix_3.html#K14" target="_self">SDSub2IntOut</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="Olive">x</font>,<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">k</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:40"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Maroon">x</font> <a href="radix_1.html#R1">is_represented_by</a> 1,<font color="Maroon">k</font> )
 ;<br/>

<a NAME="E2:40_1"/><i><font color="Green">E37</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default">1 <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="E3:40_1"/><i><font color="Green">E38</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:40_1"/>
2 <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> 1
 
;<br/>
<a NAME="E5:40_1"/><b>then </b><i><font color="Green">E41</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E6:40_1"/><i><font color="Green">E47</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default">1 <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>set </b><font color="Maroon">X</font> =  <a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,1,<font color="Maroon">k</font>;<br/>
<i><font color="Green">E48</font></i>: <span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,1,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
 <a href="radix_3.html#K12" target="_self">SDSub2INTDigit</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">x</font>,1,<font color="Maroon">k</font></span>)</span></span>)</span>,1,<font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="radix_3.html#E2"><i><font color="Green">Lemma31</font></i></a>, </i>
<br/>.= 
<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> 0</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K11" target="_self">DigB_SDSub</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,1,<font color="Maroon">k</font></span>)</span></span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K11" target="_self">DigB_SDSub</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,1,<font color="Maroon">k</font></span>)</span></span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</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">x</font>,1,<font color="Maroon">k</font></span>)</span>,1,<font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="radix_3.html#E2"><i><font color="Green">Lemma31</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">x</font>,1,<font color="Maroon">k</font></span>)</span>,1,<font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="radix_3.html#E1:40"><i><font color="Green">E30</font></i></a>, <a class="txt" href="radix_3.html#E2"><i><font color="Green">Lemma31</font></i></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">x</font>,1,<font color="Maroon">k</font></span>)</span>,1</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" 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">x</font>,1,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E3"><i><font color="Green">Lemma32</font></i></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">x</font>,1,<font color="Maroon">k</font></span>)</span>,1</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" 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">x</font>,1,<font color="Maroon">k</font></span>)</span>,0</span>)</span>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><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">x</font>,1,<font color="Maroon">k</font></span>)</span>,1</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" target="_self">SDSub_Add_Carry</a> 0,<font color="Maroon">k</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">x</font>,1,<font color="Maroon">k</font></span>)</span>,1</span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
<b>by </b><i><a class="txt" href="radix_3.html#E1:40"><i><font color="Green">E30</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,1,<font color="Maroon">k</font></span>)</span>,1</span>)</span> <a href="xcmplx_0.html#K6">-</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="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,1,<font color="Maroon">k</font></span>)</span>,1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>

;<br/>
<i><font color="Green">E49</font></i>: <span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,1,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 = 
 <a href="radix_3.html#K12" target="_self">SDSub2INTDigit</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">x</font>,1,<font color="Maroon">k</font></span>)</span></span>)</span>,2,<font color="Maroon">k</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K11" target="_self">DigB_SDSub</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,1,<font color="Maroon">k</font></span>)</span></span>)</span>,2</span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E4"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="newton.html#T10">NEWTON:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K9" target="_self">SD2SDSubDigitS</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,1,<font color="Maroon">k</font></span>)</span>,2,<font color="Maroon">k</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K8" target="_self">SD2SDSubDigit</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,1,<font color="Maroon">k</font></span>)</span>,2,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E1:40"><i><font color="Green">E30</font></i></a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</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">x</font>,1,<font color="Maroon">k</font></span>)</span>,1</span>)</span>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E4"><i><font color="Green">Lemma33</font></i></a>, , </i>
;<br/>
<b>reconsider </b><font color="Maroon">CRY</font> = <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</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">x</font>,1,<font color="Maroon">k</font></span>)</span>,1</span>)</span>,<font color="Maroon">k</font></span>)</span> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">DIG1</font> = <span class="p1">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,1,<font color="Maroon">k</font></span>)</span>,1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">CRY</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="mesfunc1.html#K2">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">DIG2</font> = <font color="Maroon">CRY</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="mesfunc1.html#K2">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<a NAME="E12:40_1"/><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,1,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E13:40_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,1,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="radix_3.html#E2"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E14:40_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,1,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,1,<font color="Maroon">k</font></span>)</span>,1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">CRY</font>
 
<b>by </b><i><a class="txt" href="radix_3.html#E1:40_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E15:40_1"/>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,1,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="radix_3.html#E2:40_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E16:40_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,1,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">CRY</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E17:40_1"/><b>then </b>
 <a href="radix_3.html#K13" target="_self">SDSub2INT</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">x</font>,1,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">DIG1</font>,<font color="Maroon">DIG2</font></span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="radix_3.html#E2:40_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="radix_3.html#E3:40_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>then </b> <a href="wsierp_1.html#K6">Sum</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,1,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> = 
<font color="Maroon">DIG1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">DIG2</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T107">RVSUM_1:107</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="radix_3.html#E1:40"><i><font color="Green">E30</font></i></a>, <a class="ref" href="radix_1.html#T24">RADIX_1:24</a></i>
;<br/>
<b>hence </b><a NAME="E19:40_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="radix_3.html#K14" target="_self">SDSub2IntOut</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">x</font>,1,<font color="Maroon">k</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:40"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><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>assume </b><a NAME="E1:40_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>] )
 ;<br/>

<a NAME="E2:40_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E3:40_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">n</font> <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#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E4:40_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <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#T5">FINSEQ_1:5</a></i>;<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> , <font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E5:40_2"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Maroon">x</font> <a href="radix_1.html#R1">is_represented_by</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">k</font> )
 ;<br/>

<b>set </b><font color="Maroon">xn</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/>
<a NAME="E6:40_2"/>
 <a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="radix_2.html#T6">RADIX_2:6</a></i>;<br/>
<a NAME="E7:40_2"/><b>then </b>
<span class="p1">(<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> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T13">PREPOWER:13</a></i>;<br/>
<a NAME="E8:40_2"/><b>then </b>
<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> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<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>
 
<b>by </b><i><a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<a NAME="E9:40_2"/><b>then </b>
<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> <a href="radix_1.html#R1">is_represented_by</a> <font color="Maroon">n</font>,<font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="radix_1.html#D12">RADIX_1:def 12</a></i>;<br/>
<b>then </b><i><font color="Green">E86</font></i>: <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> = 
 <a href="radix_3.html#K14" target="_self">SDSub2IntOut</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">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>
<b>by </b><i><a class="txt" href="radix_3.html#E2"><i><font color="Green">Lemma31</font></i></a>, </i>
<br/>.= 
 <a href="wsierp_1.html#K6">Sum</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">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">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>)</span>

;<br/>
<b>set </b><font color="Maroon">X</font> =  <a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p2">(<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>;<br/>
<b>set </b><font color="Maroon">XN</font> =  <a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">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">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>;<br/>
<a NAME="E11:40_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <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="txt" href="radix_3.html#E4"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/>
<a NAME="E12:40_2"/><i><font color="Green">E97</font></i>: 
<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 <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#T5">FINSEQ_1:5</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="mesfunc1.html#K2">INT</a>  =  <a href="radix_3.html#K12" target="_self">SDSub2INTDigit</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">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">a<sub>1</sub></font>,<font color="Maroon">k</font>;<br/>
<b>consider </b><font color="Maroon">xp</font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a> <b> such that </b><br/><a NAME="E14:40_2"/><i><font color="Green">E100</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">xp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E15:40_2"/><i><font color="Green">E101</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">xp</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon">xpp</font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a> <b> such that </b><br/><a NAME="E17:40_2"/><i><font color="Green">E103</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">xpp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E18:40_2"/><i><font color="Green">E104</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">xpp</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="mesfunc1.html#K2">INT</a>  =  <a href="radix_3.html#K12" target="_self">SDSub2INTDigit</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">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">a<sub>1</sub></font>,<font color="Maroon">k</font>;<br/>
<b>consider </b><font color="Maroon">xnpp</font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a> <b> such that </b><br/><a NAME="E20:40_2"/><i><font color="Green">E107</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">xnpp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E21:40_2"/><i><font color="Green">E108</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">xnpp</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<a NAME="E22:40_2"/><i><font color="Green">E109</font></i>: 
 <a href="radix_3.html#K13" target="_self">SDSub2INT</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">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> <a href="hidden.html#R1">=</a> <font color="Maroon">xp</font> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p3">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p5">(<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="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:40_2_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">xp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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"><span class="p5">(<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>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</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="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/>
<a NAME="E2:40_2_2"/><b>then </b><i><font color="Green">E110</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p4">(<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>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">xp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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"><span class="p5">(<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>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E3:40_2_2"/><i><font color="Green">E111</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p4">(<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>)</span> <a href="hidden.html#R1">=</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="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E4:40_2_2"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p4">(<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>)</span> holds <br/><span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p4">(<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>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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"><span class="p5">(<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>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_2_2_1"/><i><font color="Green">E112</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p4">(<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>)</span> )
 ;<br/>

<a NAME="E2:40_2_2_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</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="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E3:40_2_2_1"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">j</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/>
<a NAME="E4:40_2_2_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p4">(<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>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_2_1_1_1"/>
( <font color="Maroon">j</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> or <font color="Maroon">j</font> <a href="hidden.html#R1">=</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 )
 </span><b>by </b><i>, <a class="ref" href="finseq_2.html#T8">FINSEQ_2:8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:40_2_2_1_1_1_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">j</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>
 ;<br/><div class="add"><a NAME="E2:40_2_2_1_1_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xp</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">xp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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"><span class="p5">(<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>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">xp</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="radix_3.html#K12" target="_self">SDSub2INTDigit</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">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">j</font>,<font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="radix_3.html#E5"><i><font color="Green">Lemma39</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p4">(<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>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p4">(<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>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E4:40_2_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p4">(<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>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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"><span class="p5">(<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>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:40_2_2_1_1_1_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</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
 ;<br/><div class="add"><a NAME="E2:40_2_2_1_1_1_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p4">(<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>)</span>
 <b>by </b><i><a class="ref" href="radix_3.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">xp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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"><span class="p5">(<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>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font> = 
 <a href="radix_3.html#K12" target="_self">SDSub2INTDigit</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">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"><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>,<font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p4">(<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>)</span> <a href="finseq_4.html#K4">/.</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>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p4">(<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>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="radix_3.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E4:40_2_2_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p4">(<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>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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"><span class="p5">(<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>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:40_2_2_1"/>
<span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p4">(<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>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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"><span class="p5">(<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>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:40_2_2"/>
 <a href="radix_3.html#K13" target="_self">SDSub2INT</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">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> <a href="hidden.html#R1">=</a> <font color="Maroon">xp</font> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p3">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p5">(<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="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:40_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">xp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xpp</font> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p3">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p5">(<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="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><a href="binarith.html#K11">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:40_2_3"/><i><font color="Green">E119</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">xp</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">xpp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/>
<a NAME="E2:40_2_3"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">xp</font> holds <br/><font color="Maroon">xp</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xpp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_2_3_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">xp</font> )
 ;<br/>

<a NAME="E2:40_2_3_1"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">j</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_3_1_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="finseq_2.html#T8">FINSEQ_2:8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:40_2_3_1_1_1_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:40_2_3_1_1_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xpp</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">xpp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">xpp</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="radix_3.html#K12" target="_self">SDSub2INTDigit</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">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">j</font>,<font color="Maroon">k</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">xp</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="radix_3.html#E5"><i><font color="Green">Lemma39</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E4:40_2_3_1_1_1_1"/>
<font color="Maroon">xp</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xpp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:40_2_3_1_1_1_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><font color="Maroon">xpp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font> = 
 <a href="radix_3.html#K12" target="_self">SDSub2INTDigit</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">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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
<br/>.= 
<font color="Maroon">xp</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="radix_3.html#E5"><i><font color="Green">Lemma39</font></i></a>, , </i>
;<br/><b>hence </b><a NAME="E3:40_2_3_1_1_1_2"/>
<font color="Maroon">xp</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xpp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:40_2_3_1"/>
<font color="Maroon">xp</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xpp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:40_2_3"/>
<font color="Maroon">xp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xpp</font> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p3">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p5">(<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="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><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="radix_3.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:40_2"/><i><font color="Green">E123</font></i>: 
 <a href="radix_3.html#K13" target="_self">SDSub2INT</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">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> <a href="hidden.html#R1">=</a> <font color="Maroon">xnpp</font> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p3">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p0">(<span class="default"><span class="p1">(<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="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><a href="binarith.html#K11">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:40_2_4"/><i><font color="Green">E124</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">xnpp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/>
<a NAME="E2:40_2_4"/><i><font color="Green">E125</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">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">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>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E3:40_2_4"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">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">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>)</span> holds <br/><span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">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">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>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xnpp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_2_4_1"/><i><font color="Green">E126</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">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">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>)</span> )
 ;<br/>

<a NAME="E2:40_2_4_1"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">j</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="E3:40_2_4_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">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">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>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_4_1_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="finseq_2.html#T8">FINSEQ_2:8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:40_2_4_1_1_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:40_2_4_1_1_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xnpp</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">xnpp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">xnpp</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="radix_3.html#K12" target="_self">SDSub2INTDigit</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">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">j</font>,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="radix_3.html#T2" target="_self">Th2</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">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">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>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">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">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>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="radix_3.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E4:40_2_4_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">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">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>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xnpp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:40_2_4_1_1_1_2"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:40_2_4_1_1_1_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">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">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>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">xnpp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font> = 
 <a href="radix_3.html#K12" target="_self">SDSub2INTDigit</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">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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">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">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>)</span> <a href="finseq_4.html#K4">/.</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>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">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">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>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E4:40_2_4_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">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">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>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xnpp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:40_2_4_1"/>
<span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">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">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>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xnpp</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" target="_self">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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:40_2_4"/>
 <a href="radix_3.html#K13" target="_self">SDSub2INT</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">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> <a href="hidden.html#R1">=</a> <font color="Maroon">xnpp</font> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p3">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p0">(<span class="default"><span class="p1">(<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="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><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:40_2"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">xpp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xnpp</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_5"><b>proof </b></a><div class="add">

<a NAME="E1:40_2_5"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">xnpp</font> holds <br/><font color="Maroon">xnpp</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xpp</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_5_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_2_5_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">xnpp</font> )
 ;<br/>

<a NAME="E2:40_2_5_1"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">j</font> <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/>
<b>then </b><font color="Maroon">xpp</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font> = 
 <a href="radix_3.html#K12" target="_self">SDSub2INTDigit</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">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">j</font>,<font color="Maroon">k</font>
<b>by </b><i/>
<br/>.= 
 <a href="radix_3.html#K12" target="_self">SDSub2INTDigit</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">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">j</font>,<font color="Maroon">k</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">xnpp</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="radix_3.html#T2" target="_self">Th2</a>, </i>
;<br/>
<b>hence </b><a NAME="E4:40_2_5_1"/>
<font color="Maroon">xnpp</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xpp</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:40_2_5"/>
<font color="Maroon">xpp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xnpp</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E134</font></i>:  <a href="wsierp_1.html#K6">Sum</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p4">(<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>)</span> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6">Sum</a> <font color="Maroon">xp</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p4">(<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="p2">(<span class="default"><span class="p3">(<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>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="radix_3.html#T3" target="_self">Th3</a>, <a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K6">Sum</a> <font color="Maroon">xnpp</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p3">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p5">(<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="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> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p4">(<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="p2">(<span class="default"><span class="p3">(<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>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>
;<br/>
<a NAME="E27:40_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<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> <a href="nat_1.html#K1">+</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6">Sum</a> <font color="Maroon">xnpp</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">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">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="p2">(<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>
 
<b>by </b><i>, <a class="ref" href="radix_3.html#T4" target="_self">Th4</a>, <a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>;<br/>
<a NAME="E28:40_2"/><b>then </b><i><font color="Green">E135</font></i>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6">Sum</a> <font color="Maroon">xnpp</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<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> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="radix_3.html#K12" target="_self">SDSub2INTDigit</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">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">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="p2">(<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>
 
;<br/>
<b>set </b><font color="Maroon">RN</font> = <span class="p1">(<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>;<br/>
<b>set </b><font color="Maroon">RN1</font> = <span class="p1">(<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> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<i><font color="Green">E138</font></i>:  <a href="radix_3.html#K12" target="_self">SDSub2INTDigit</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">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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font> = 
<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> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K11" target="_self">DigB_SDSub</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">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">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="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<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> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K9" target="_self">SD2SDSubDigitS</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 class="p2">(<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>
<b>by </b><i><a class="txt" href="radix_3.html#E4"><i><font color="Green">Lemma33</font></i></a>, </i>
<br/>.= 
<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> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K8" target="_self">SD2SDSubDigit</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 class="p2">(<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>
<b>by </b><i><a class="txt" href="radix_3.html#E4"><i><font color="Green">Lemma33</font></i></a>, , </i>
<br/>.= 
<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> <a href="xcmplx_0.html#K3">*</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">x</font> <a href="nat_1.html#K6">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">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 class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<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> <a href="xcmplx_0.html#K3">*</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">x</font> <a href="nat_1.html#K6">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">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>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<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> <a href="xcmplx_0.html#K3">*</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">x</font>,<span class="p4">(<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>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E3"><i><font color="Green">Lemma32</font></i></a>, </i>
;<br/>
<i><font color="Green">E139</font></i>:  <a href="radix_3.html#K12" target="_self">SDSub2INTDigit</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">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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font> = 
<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> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K11" target="_self">DigB_SDSub</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p4">(<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="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<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> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K9" target="_self">SD2SDSubDigitS</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 class="p2">(<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>
<b>by </b><i>, </i>
<br/>.= 
<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> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K8" target="_self">SD2SDSubDigit</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 class="p2">(<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>
<b>by </b><i>, , </i>
<br/>.= 
<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> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K6" target="_self">SDSub_Add_Data</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p5">(<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 class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p5">(<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 class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E4"><i><font color="Green">Lemma33</font></i></a>, </i>
<br/>.= 
<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> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K6" target="_self">SDSub_Add_Data</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p5">(<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 class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p5">(<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>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p5">(<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 class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><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> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p4">(<span class="default"><a href="radix_1.html#K4">DigA</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 class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p5">(<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>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p5">(<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 class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p4">(<span class="default"><a href="radix_1.html#K4">DigA</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 class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p5">(<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>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/>
<i><font color="Green">E140</font></i>:  <a href="radix_3.html#K12" target="_self">SDSub2INTDigit</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">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"><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>,<font color="Maroon">k</font> = 
<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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K11" target="_self">DigB_SDSub</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p4">(<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="p2">(<span class="default"><span class="p3">(<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></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K9" target="_self">SD2SDSubDigitS</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 class="p2">(<span class="default"><span class="p3">(<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>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K8" target="_self">SD2SDSubDigit</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 class="p2">(<span class="default"><span class="p3">(<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>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</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">x</font>,<span class="p4">(<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 class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</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">x</font>,<span class="p4">(<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 class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/>
<b>set </b><font color="Maroon">RNSDC</font> = <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> <a href="xcmplx_0.html#K3">*</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">x</font>,<span class="p4">(<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>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">RNDIG</font> = <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> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4">DigA</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 class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">RN1SDC</font> = <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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</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">x</font>,<span class="p4">(<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 class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>;<br/>
<i><font color="Green">E144</font></i>:  <a href="wsierp_1.html#K6">Sum</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" target="_self">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" target="_self">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p4">(<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>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><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> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K4">DigA</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 class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p4">(<span class="default"><a href="radix_1.html#K4">DigA</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>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">x</font>,<span class="p5">(<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>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="radix_3.html#T6" target="_self">Th6</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<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> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><span class="p4">(<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></span>)</span>
<b>by </b><i>, <a class="ref" href="radix_1.html#T27">RADIX_1:27</a></i>
;<br/>
<a NAME="E33:40_2"/>
 <a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="radix_2.html#T6">RADIX_2:6</a></i>;<br/>
<a NAME="E34:40_2"/><b>then </b>
<span class="p1">(<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> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T13">PREPOWER:13</a></i>;<br/>
<a NAME="E35:40_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><span class="p4">(<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> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<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> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<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>
 
<b>by </b><i><a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/>
<b>hence </b><a NAME="E36:40_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="radix_3.html#K14" target="_self">SDSub2IntOut</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">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>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:40"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S9">NAT_1:sch 9</a>(<a class="txt" href="radix_3.html#E1"><i><font color="Green">Lemma29</font></i></a>, <a class="txt" href="radix_3.html#E1:40"><i><font color="Green">E30</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:40"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/>for <font color="Olive">k</font>, <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Olive">x</font> <a href="radix_1.html#R1">is_represented_by</a> <font color="Olive">n</font>,<font color="Olive">k</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="radix_3.html#K14" target="_self">SDSub2IntOut</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="Olive">x</font>,<font color="Olive">n</font>,<font color="Olive">k</font></span>)</span></span>)</span>
 ;<br/>


</div>
