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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Olive" title="b2">x</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> $1,<font color="Olive" title="b1">k</font> holds <br/><font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_3.html#K14" title="RADIX_3:func.14">SDSub2IntOut</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Olive" title="b2">x</font>,$1,<font color="Olive" title="b1">k</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:40"/><i><font color="Green" title="E29">A1</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" title="c1">k</font>, <font color="Maroon" title="c2">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:40_1"/><i><font color="Green" title="E30">A2</font></i>: 
( <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c2">x</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> 1,<font color="Maroon" title="c1">k</font> )
 ;<br/>

<a NAME="E2:40_1"/><i><font color="Green" title="E31">A3</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:40_1"/><i><font color="Green" title="E32">A4</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:40_1"/>
2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
;<br/>
<a NAME="E5:40_1"/><b>then </b><i><font color="Green" title="E33">A5</font></i>: 
2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E6:40_1"/><i><font color="Green" title="E34">A6</font></i>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">X</font> =  <a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font>;<br/>
<i><font color="Green" title="E35">A7</font></i>: <span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
 <a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span>,1,<font color="Maroon" title="c1">k</font>
<b>by </b><i><a class="txt" href="radix_3.html#E2:40_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="radix_3.html#D11" target="_self" title="RADIX_3:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 0</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K11" title="RADIX_3:func.11">DigB_SDSub</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K11" title="RADIX_3:func.11">DigB_SDSub</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>
<br/>.= 
 <a href="radix_3.html#K9" title="RADIX_3:func.9">SD2SDSubDigitS</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,1,<font color="Maroon" title="c1">k</font>
<b>by </b><i><a class="txt" href="radix_3.html#E2:40_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="radix_3.html#D8" target="_self" title="RADIX_3:def.8">Def8</a></i>
<br/>.= 
 <a href="radix_3.html#K8" title="RADIX_3:func.8">SD2SDSubDigit</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,1,<font color="Maroon" title="c1">k</font>
<b>by </b><i><a class="txt" href="radix_3.html#E1:40_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="radix_3.html#E2:40_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="radix_3.html#D7" target="_self" title="RADIX_3:def.7">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,1</span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,<span class="p3">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E3:40_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="radix_3.html#D6" target="_self" title="RADIX_3:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,1</span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,0</span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,1</span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> 0,<font color="Maroon" title="c1">k</font></span>)</span>
<b>by </b><i><a class="ref" href="radix_1.html#D3" title="RADIX_1:def.3">RADIX_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,1</span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0
<b>by </b><i><a class="txt" href="radix_3.html#E1:40_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="radix_3.html#T17" target="_self" title="RADIX_3:th.17">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,1</span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span>

;<br/>
<i><font color="Green" title="E36">A8</font></i>: <span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2 = 
 <a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span>,2,<font color="Maroon" title="c1">k</font>
<b>by </b><i><a class="txt" href="radix_3.html#E6:40_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="radix_3.html#D11" target="_self" title="RADIX_3:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K11" title="RADIX_3:func.11">DigB_SDSub</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span>,2</span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E5:40_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K9" title="RADIX_3:func.9">SD2SDSubDigitS</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,2,<font color="Maroon" title="c1">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E6:40_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="radix_3.html#D8" target="_self" title="RADIX_3:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K8" title="RADIX_3:func.8">SD2SDSubDigit</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,2,<font color="Maroon" title="c1">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E1:40_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="radix_3.html#E6:40_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="radix_3.html#D7" target="_self" title="RADIX_3:def.7">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,1</span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E5:40_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="radix_3.html#E6:40_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="radix_3.html#D6" target="_self" title="RADIX_3:def.6">Def6</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c4">CRY</font> = <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,1</span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c5">DIG1</font> = <span class="p1">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">CRY</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">DIG2</font> = <font color="Maroon" title="c4">CRY</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<a NAME="E12:40_1"/><i><font color="Green" title="E37">A9</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E13:40_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">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" title="E31">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E14:40_1"/><b>then </b><i><font color="Green" title="E38">A10</font></i>: 
<span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">CRY</font>
 
<b>by </b><i><a class="txt" href="radix_3.html#E7:40_1"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E15:40_1"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="radix_3.html#E6:40_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="radix_3.html#E12:40_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">CRY</font>
 
<b>by </b><i><a class="txt" href="radix_3.html#E8:40_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E17:40_1"/><b>then </b>
 <a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">DIG1</font>,<font color="Maroon" title="c6">DIG2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="radix_3.html#E12:40_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="txt" href="radix_3.html#E14:40_1"><i><font color="Green" title="E38">A10</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<b>then </b> <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span> = 
<font color="Maroon" title="c5">DIG1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">DIG2</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T107" title="RVSUM_1:th.107">RVSUM_1:107</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="txt" href="radix_3.html#E1:40_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="radix_1.html#T24" title="RADIX_1:th.24">RADIX_1:24</a></i>
;<br/>
<b>hence </b><a NAME="E19:40_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_3.html#K14" title="RADIX_3:func.14">SDSub2IntOut</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c2">x</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:40"/><i><font color="Green" title="E30">A11</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:40_2"/><i><font color="Green" title="E31">A12</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>] )
 ;<br/>

<a NAME="E2:40_2"/><b>then </b><i><font color="Green" title="E32">A13</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E3:40_2"/><i><font color="Green" title="E33">A14</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<b>let </b><font color="Maroon" title="c2">k</font>, <font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E4:40_2"/><i><font color="Green" title="E34">A15</font></i>: 
( <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c3">x</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c2">k</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c4">xn</font> = <font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>
<a NAME="E5:40_2"/>
 <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="radix_2.html#T6" title="RADIX_2:th.6">RADIX_2:6</a></i>;<br/>
<a NAME="E6:40_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T13" title="PREPOWER:th.13">PREPOWER:13</a></i>;<br/>
<a NAME="E7:40_2"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></i>;<br/>
<a NAME="E8:40_2"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font>
 
<b>by </b><i><a class="ref" href="radix_1.html#D12" title="RADIX_1:def.12">RADIX_1:def 12</a></i>;<br/>
<b>then </b><i><font color="Green" title="E35">A16</font></i>: <font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> = 
 <a href="radix_3.html#K14" title="RADIX_3:func.14">SDSub2IntOut</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E1:40_2"><i><font color="Green" title="E31">A12</font></i></a>, <a class="txt" href="radix_3.html#E4:40_2"><i><font color="Green" title="E34">A15</font></i></a></i>
<br/>.= 
 <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span>

;<br/>
<b>set </b><font color="Maroon" title="c5">X</font> =  <a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">XN</font> =  <a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>;<br/>
<a NAME="E10:40_2"/><i><font color="Green" title="E36">A17</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="radix_3.html#E3:40_2"><i><font color="Green" title="E33">A14</font></i></a>, <a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></i>;<br/>
<a NAME="E11:40_2"/><i><font color="Green" title="E37">A18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  =  <a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,$1,<font color="Maroon" title="c2">k</font>;<br/>
<b>consider </b><font color="Maroon" title="c7">xp</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E13:40_2"/><i><font color="Green" title="E38">A19</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">xp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>and </b><br/><a NAME="E14:40_2"/><i><font color="Green" title="E39">A20</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c7">xp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon" title="c8">xpp</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E16:40_2"/><i><font color="Green" title="E40">A21</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">xpp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>and </b><br/><a NAME="E17:40_2"/><i><font color="Green" title="E41">A22</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c8">xpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  =  <a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,$1,<font color="Maroon" title="c2">k</font>;<br/>
<b>consider </b><font color="Maroon" title="c9">xnpp</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E19:40_2"/><i><font color="Green" title="E42">A23</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">xnpp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>and </b><br/><a NAME="E20:40_2"/><i><font color="Green" title="E43">A24</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c9">xnpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i>
<a NAME="E21:40_2"/><i><font color="Green" title="E44">A25</font></i>: 
 <a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">xp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p3">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&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" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="radix_3.html#E13:40_2"><i><font color="Green" title="E38">A19</font></i></a>, <a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>;<br/>
<a NAME="E2:40_2_2"/><b>then </b><i><font color="Green" title="E45">A26</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E3:40_2_2"/><i><font color="Green" title="E46">A27</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E4:40_2_2"/>
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c10">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:40_2_2_1"/><i><font color="Green" title="E47">A28</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> &amp; <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:40_2_2_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> &amp; <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E3:40_2_2_1"/><b>then </b><i><font color="Green" title="E48">A29</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:40_2_2_1"/><i><font color="Green" title="E49">A30</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="radix_3.html#E1:40_2_2_1"><i><font color="Green" title="E47">A28</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> or <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="radix_3.html#E3:40_2_2_1"><i><font color="Green" title="E48">A29</font></i></a>, <a class="ref" href="finseq_2.html#T8" title="FINSEQ_2:th.8">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" title="E50">A31</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:40_2_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">xp</font>
 <b>by </b><i><a class="txt" href="radix_3.html#E13:40_2"><i><font color="Green" title="E38">A19</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c7">xp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> = 
<font color="Maroon" title="c7">xp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c2">k</font>
<b>by </b><i><a class="txt" href="radix_3.html#E14:40_2"><i><font color="Green" title="E39">A20</font></i></a>, <a class="txt" href="radix_3.html#E1:40_2_2_1_1_1_1"><i><font color="Green" title="E50">A31</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="txt" href="radix_3.html#E3:40_2_2_1"><i><font color="Green" title="E48">A29</font></i></a>, <a class="ref" href="radix_3.html#D11" target="_self" title="RADIX_3:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="txt" href="radix_3.html#E4:40_2_2_1"><i><font color="Green" title="E49">A30</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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" title="E50">A32</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:40_2_2_1_1_1_2"/><i><font color="Green" title="E51">A33</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="radix_3.html#E3:40_2_2"><i><font color="Green" title="E46">A27</font></i></a>, <a class="txt" href="radix_3.html#E3:40_2_2_1"><i><font color="Green" title="E48">A29</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">xp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> = 
 <a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font>
<b>by </b><i><a class="txt" href="radix_3.html#E13:40_2"><i><font color="Green" title="E38">A19</font></i></a>, <a class="txt" href="radix_3.html#E1:40_2_2_1_1_1_2"><i><font color="Green" title="E50">A32</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E3:40_2_2_1"><i><font color="Green" title="E48">A29</font></i></a>, <a class="txt" href="radix_3.html#E1:40_2_2_1_1_1_2"><i><font color="Green" title="E50">A32</font></i></a>, <a class="ref" href="radix_3.html#D11" target="_self" title="RADIX_3:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="txt" href="radix_3.html#E1:40_2_2_1_1_1_2"><i><font color="Green" title="E50">A32</font></i></a>, <a class="txt" href="radix_3.html#E2:40_2_2_1_1_1_2"><i><font color="Green" title="E51">A33</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:40_2_2"/>
 <a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">xp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p3">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="radix_3.html#E2:40_2_2"><i><font color="Green" title="E45">A26</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:40_2"/><i><font color="Green" title="E45">A34</font></i>: 
<font color="Maroon" title="c7">xp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">xpp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p3">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&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" title="E46">A35</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">xp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">xpp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="radix_3.html#E13:40_2"><i><font color="Green" title="E38">A19</font></i></a>, <a class="txt" href="radix_3.html#E16:40_2"><i><font color="Green" title="E40">A21</font></i></a>, <a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>;<br/>
<a NAME="E2:40_2_3"/>
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">xp</font> holds <br/><font color="Maroon" title="c7">xp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">xpp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c10">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:40_2_3_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> &amp; <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">xp</font> )
 ;<br/>

<a NAME="E2:40_2_3_1"/><b>then </b><i><font color="Green" title="E47">A36</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="radix_3.html#E13:40_2"><i><font color="Green" title="E38">A19</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="radix_3.html#E2:40_2_3_1"><i><font color="Green" title="E47">A36</font></i></a>, <a class="ref" href="finseq_2.html#T8" title="FINSEQ_2:th.8">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" title="E48">A37</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><a NAME="E2:40_2_3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">xpp</font>
 <b>by </b><i><a class="txt" href="radix_3.html#E16:40_2"><i><font color="Green" title="E40">A21</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c8">xpp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> = 
<font color="Maroon" title="c8">xpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c2">k</font>
<b>by </b><i><a class="txt" href="radix_3.html#E17:40_2"><i><font color="Green" title="E41">A22</font></i></a>, <a class="txt" href="radix_3.html#E1:40_2_3_1_1_1_1"><i><font color="Green" title="E48">A37</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">xp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="txt" href="radix_3.html#E14:40_2"><i><font color="Green" title="E39">A20</font></i></a>, <a class="txt" href="radix_3.html#E2:40_2_3_1"><i><font color="Green" title="E47">A36</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:40_2_3_1_1_1_1"/>
<font color="Maroon" title="c7">xp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">xpp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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" title="E48">A38</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c8">xpp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> = 
 <a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font>
<b>by </b><i><a class="txt" href="radix_3.html#E16:40_2"><i><font color="Green" title="E40">A21</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>
<br/>.= 
<font color="Maroon" title="c7">xp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="txt" href="radix_3.html#E14:40_2"><i><font color="Green" title="E39">A20</font></i></a>, <a class="txt" href="radix_3.html#E2:40_2_3_1"><i><font color="Green" title="E47">A36</font></i></a>, <a class="txt" href="radix_3.html#E1:40_2_3_1_1_1_2"><i><font color="Green" title="E48">A38</font></i></a></i>
;<br/><b>hence </b><a NAME="E3:40_2_3_1_1_1_2"/>
<font color="Maroon" title="c7">xp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">xpp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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" title="c7">xp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">xpp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:40_2_3"/>
<font color="Maroon" title="c7">xp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">xpp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p3">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="radix_3.html#E1:40_2_3"><i><font color="Green" title="E46">A35</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:40_2"/><i><font color="Green" title="E46">A39</font></i>: 
 <a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">xnpp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p3">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&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" title="E47">A40</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">xnpp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="radix_3.html#E19:40_2"><i><font color="Green" title="E42">A23</font></i></a>, <a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>;<br/>
<a NAME="E2:40_2_4"/><i><font color="Green" title="E48">A41</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E3:40_2_4"/>
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">xnpp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c10">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:40_2_4_1"/><i><font color="Green" title="E49">A42</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> &amp; <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:40_2_4_1"/><b>then </b><i><font color="Green" title="E50">A43</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="radix_3.html#E2:40_2_4"><i><font color="Green" title="E48">A41</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:40_2_4_1"/><i><font color="Green" title="E51">A44</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="radix_3.html#E1:40_2_4_1"><i><font color="Green" title="E49">A42</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="radix_3.html#E2:40_2_4_1"><i><font color="Green" title="E50">A43</font></i></a>, <a class="ref" href="finseq_2.html#T8" title="FINSEQ_2:th.8">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" title="E52">A45</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><a NAME="E2:40_2_4_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">xnpp</font>
 <b>by </b><i><a class="txt" href="radix_3.html#E19:40_2"><i><font color="Green" title="E42">A23</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c9">xnpp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> = 
<font color="Maroon" title="c9">xnpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c2">k</font>
<b>by </b><i><a class="txt" href="radix_3.html#E20:40_2"><i><font color="Green" title="E43">A24</font></i></a>, <a class="txt" href="radix_3.html#E1:40_2_4_1_1_1_1"><i><font color="Green" title="E52">A45</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="txt" href="radix_3.html#E2:40_2_4_1"><i><font color="Green" title="E50">A43</font></i></a>, <a class="ref" href="radix_3.html#D11" target="_self" title="RADIX_3:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="txt" href="radix_3.html#E3:40_2_4_1"><i><font color="Green" title="E51">A44</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">xnpp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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" title="E52">A46</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:40_2_4_1_1_1_2"/><i><font color="Green" title="E53">A47</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="radix_3.html#E2:40_2_4"><i><font color="Green" title="E48">A41</font></i></a>, <a class="txt" href="radix_3.html#E2:40_2_4_1"><i><font color="Green" title="E50">A43</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">xnpp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> = 
 <a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font>
<b>by </b><i><a class="txt" href="radix_3.html#E19:40_2"><i><font color="Green" title="E42">A23</font></i></a>, <a class="txt" href="radix_3.html#E1:40_2_4_1_1_1_2"><i><font color="Green" title="E52">A46</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E2:40_2_4_1"><i><font color="Green" title="E50">A43</font></i></a>, <a class="txt" href="radix_3.html#E1:40_2_4_1_1_1_2"><i><font color="Green" title="E52">A46</font></i></a>, <a class="ref" href="radix_3.html#D11" target="_self" title="RADIX_3:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="txt" href="radix_3.html#E1:40_2_4_1_1_1_2"><i><font color="Green" title="E52">A46</font></i></a>, <a class="txt" href="radix_3.html#E2:40_2_4_1_1_1_2"><i><font color="Green" title="E53">A47</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">xnpp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">xnpp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p4">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:40_2_4"/>
 <a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">xnpp</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p3">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="radix_3.html#E1:40_2_4"><i><font color="Green" title="E47">A40</font></i></a>, <a class="txt" href="radix_3.html#E2:40_2_4"><i><font color="Green" title="E48">A41</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:40_2"/><i><font color="Green" title="E47">A48</font></i>: 
<font color="Maroon" title="c8">xpp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">xnpp</font> holds <br/><font color="Maroon" title="c9">xnpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">xpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c10">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:40_2_5_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> &amp; <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">xnpp</font> )
 ;<br/>

<a NAME="E2:40_2_5_1"/><b>then </b><i><font color="Green" title="E48">A49</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="radix_3.html#E19:40_2"><i><font color="Green" title="E42">A23</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>then </b><font color="Maroon" title="c8">xpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> = 
 <a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c2">k</font>
<b>by </b><i><a class="txt" href="radix_3.html#E17:40_2"><i><font color="Green" title="E41">A22</font></i></a></i>
<br/>.= 
 <a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c2">k</font>
<b>by </b><i><a class="txt" href="radix_3.html#E4:40_2"><i><font color="Green" title="E34">A15</font></i></a>, <a class="txt" href="radix_3.html#E2:40_2_5_1"><i><font color="Green" title="E48">A49</font></i></a>, <a class="ref" href="radix_3.html#T21" target="_self" title="RADIX_3:th.21">Th21</a></i>
<br/>.= 
<font color="Maroon" title="c9">xnpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="txt" href="radix_3.html#E20:40_2"><i><font color="Green" title="E43">A24</font></i></a>, <a class="txt" href="radix_3.html#E2:40_2_5_1"><i><font color="Green" title="E48">A49</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E4:40_2_5_1"/>
<font color="Maroon" title="c9">xnpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">xpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:40_2_5"/>
<font color="Maroon" title="c8">xpp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">xnpp</font>
 <b>by </b><i><a class="txt" href="radix_3.html#E16:40_2"><i><font color="Green" title="E40">A21</font></i></a>, <a class="txt" href="radix_3.html#E19:40_2"><i><font color="Green" title="E42">A23</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green" title="E48">A50</font></i>:  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <font color="Maroon" title="c7">xp</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E21:40_2"><i><font color="Green" title="E44">A25</font></i></a>, <a class="ref" href="rvsum_1.html#T104" title="RVSUM_1:th.104">RVSUM_1:104</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <font color="Maroon" title="c9">xnpp</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p3">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E22:40_2"><i><font color="Green" title="E45">A34</font></i></a>, <a class="txt" href="radix_3.html#E24:40_2"><i><font color="Green" title="E47">A48</font></i></a>, <a class="ref" href="rvsum_1.html#T104" title="RVSUM_1:th.104">RVSUM_1:104</a></i>
;<br/>
<a NAME="E26:40_2"/><i><font color="Green" title="E49">A51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <font color="Maroon" title="c9">xnpp</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="radix_3.html#E9:40_2"><i><font color="Green" title="E35">A16</font></i></a>, <a class="txt" href="radix_3.html#E23:40_2"><i><font color="Green" title="E46">A39</font></i></a>, <a class="ref" href="rvsum_1.html#T104" title="RVSUM_1:th.104">RVSUM_1:104</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">RN</font> = <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font>;<br/>
<b>set </b><font color="Maroon" title="c11">RN1</font> = <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<i><font color="Green" title="E50">A52</font></i>:  <a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K11" title="RADIX_3:func.11">DigB_SDSub</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K9" title="RADIX_3:func.9">SD2SDSubDigitS</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E3:40_2"><i><font color="Green" title="E33">A14</font></i></a>, <a class="ref" href="radix_3.html#D8" target="_self" title="RADIX_3:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K8" title="RADIX_3:func.8">SD2SDSubDigit</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E3:40_2"><i><font color="Green" title="E33">A14</font></i></a>, <a class="txt" href="radix_3.html#E4:40_2"><i><font color="Green" title="E34">A15</font></i></a>, <a class="ref" href="radix_3.html#D7" target="_self" title="RADIX_3:def.7">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>
<b>by </b><i><a class="ref" href="radix_3.html#D6" target="_self" title="RADIX_3:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E2:40_2"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="radix_3.html#E5"><i><font color="Green" title="E5">Lm5</font></i></a></i>
;<br/>
<i><font color="Green" title="E51">A53</font></i>:  <a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K11" title="RADIX_3:func.11">DigB_SDSub</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K9" title="RADIX_3:func.9">SD2SDSubDigitS</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E10:40_2"><i><font color="Green" title="E36">A17</font></i></a>, <a class="ref" href="radix_3.html#D8" target="_self" title="RADIX_3:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K8" title="RADIX_3:func.8">SD2SDSubDigit</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E4:40_2"><i><font color="Green" title="E34">A15</font></i></a>, <a class="txt" href="radix_3.html#E10:40_2"><i><font color="Green" title="E36">A17</font></i></a>, <a class="ref" href="radix_3.html#D7" target="_self" title="RADIX_3:def.7">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E3:40_2"><i><font color="Green" title="E33">A14</font></i></a>, <a class="ref" href="radix_3.html#D6" target="_self" title="RADIX_3:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">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" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p4">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">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" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p4">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
;<br/>
<i><font color="Green" title="E52">A54</font></i>:  <a href="radix_3.html#K12" title="RADIX_3:func.12">SDSub2INTDigit</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K11" title="RADIX_3:func.11">DigB_SDSub</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K9" title="RADIX_3:func.9">SD2SDSubDigitS</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E11:40_2"><i><font color="Green" title="E37">A18</font></i></a>, <a class="ref" href="radix_3.html#D8" target="_self" title="RADIX_3:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K8" title="RADIX_3:func.8">SD2SDSubDigit</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E4:40_2"><i><font color="Green" title="E34">A15</font></i></a>, <a class="txt" href="radix_3.html#E11:40_2"><i><font color="Green" title="E37">A18</font></i></a>, <a class="ref" href="radix_3.html#D7" target="_self" title="RADIX_3:def.7">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>
<b>by </b><i><a class="ref" href="radix_3.html#D6" target="_self" title="RADIX_3:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c12">RNSDC</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c13">RNDIG</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c14">RN1SDC</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>;<br/>
<i><font color="Green" title="E53">A55</font></i>:  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p1">(<span class="default"><a href="radix_3.html#K13" title="RADIX_3:func.13">SDSub2INT</a> <span class="p2">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p4">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p5">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E25:40_2"><i><font color="Green" title="E48">A50</font></i></a>, <a class="txt" href="radix_3.html#E26:40_2"><i><font color="Green" title="E49">A51</font></i></a>, <a class="txt" href="radix_3.html#E27:40_2"><i><font color="Green" title="E50">A52</font></i></a>, <a class="txt" href="radix_3.html#E28:40_2"><i><font color="Green" title="E51">A53</font></i></a>, <a class="txt" href="radix_3.html#E29:40_2"><i><font color="Green" title="E52">A54</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="radix_3.html#E4:40_2"><i><font color="Green" title="E34">A15</font></i></a>, <a class="ref" href="radix_1.html#T27" title="RADIX_1:th.27">RADIX_1:27</a></i>
;<br/>
<a NAME="E31:40_2"/>
 <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="radix_2.html#T6" title="RADIX_2:th.6">RADIX_2:6</a></i>;<br/>
<a NAME="E32:40_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T13" title="PREPOWER:th.13">PREPOWER:13</a></i>;<br/>
<b>hence </b><a NAME="E33:40_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_3.html#K14" title="RADIX_3:func.14">SDSub2IntOut</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="radix_3.html#E30:40_2"><i><font color="Green" title="E53">A55</font></i></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:40"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S8" title="NAT_1:sch.8">NAT_1:sch 8</a>(<a class="txt" href="radix_3.html#E1:40"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="radix_3.html#E2:40"><i><font color="Green" title="E30">A11</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:40"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/>for <font color="Olive" title="b2">k</font>, <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Olive" title="b3">x</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">k</font> holds <br/><font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_3.html#K14" title="RADIX_3:func.14">SDSub2IntOut</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">k</font></span>)</span></span>)</span>
 ;<br/>


</div>
