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

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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="radix_1.html#R1" target="_self">is_represented_by</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">k</font> holds <br/><font color="Olive">m</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K8" target="_self">SDDec</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <font color="Olive">m</font>,<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">k</font></span>)</span>;<br/>
<a NAME="E1:35"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:35_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">m</font> <a href="radix_1.html#R1" target="_self">is_represented_by</a> 1,<font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:35_1"/><i><font color="Green">E37</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
 <a href="radix_1.html#K6" target="_self">SubDigit</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <font color="Maroon">m</font>,1,<font color="Maroon">k</font></span>)</span>,1,<font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> 0</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K5" target="_self">DigB</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <font color="Maroon">m</font>,1,<font color="Maroon">k</font></span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K5" target="_self">DigB</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <font color="Maroon">m</font>,1,<font color="Maroon">k</font></span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i>, </i>
;<br/>
<a NAME="E4:35_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <font color="Maroon">m</font>,1,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="radix_1.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E5:35_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="mesfunc1.html#K2">INT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">M</font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">m</font></span><a href="finseq_1.html#K9">*&gt;</a></span> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T95">FINSEQ_1:95</a></i>;<br/>
<b>thus </b> <a href="radix_1.html#K8" target="_self">SDDec</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <font color="Maroon">m</font>,1,<font color="Maroon">k</font></span>)</span> = 
 <a href="wsierp_1.html#K6">Sum</a> <font color="Maroon">M</font>
<b>by </b><i><a class="txt" href="radix_1.html#E1:35"><i><font color="Green">E29</font></i></a>, </i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i><a class="txt" href="radix_1.html#E1:35_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:35"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">u</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">u</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">u</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:35_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">u</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">u</font>] )
 ;<br/>

<a NAME="E2:35_2"/>
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="radix_1.html#R1" target="_self">is_represented_by</a> <font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">k</font> holds <br/><font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K8" target="_self">SDDec</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <font color="Olive">p</font>,<span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:35_2_1"/>
<font color="Maroon">p</font> <a href="radix_1.html#R1" target="_self">is_represented_by</a> <font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:35_2_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">R</font> = <span class="p1">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font>;<br/>
<a NAME="E3:35_2_1"/>
 <a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E4:35_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T12">PREPOWER:12</a></i>;<br/>
<a NAME="E5:35_2_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E6:35_2_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</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" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/>
<b>set </b><font color="Maroon">M</font> = <font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">N</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>;<br/>
<a NAME="E7:35_2_1"/>
<font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<a NAME="E8:35_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span> <a href="radix_1.html#R1" target="_self">is_represented_by</a> <font color="Maroon">u</font>,<font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:35_2_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8" target="_self">SDDec</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">u</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="radix_1.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E10:35_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="mesfunc1.html#K2">INT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">NN</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><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" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T95">FINSEQ_1:95</a></i>;<br/>
<b>reconsider </b><font color="Maroon">DD</font> =  <a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">u</font>,<font color="Maroon">k</font></span>)</span> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a>  ;<br/>
<a NAME="E13:35_2_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">DD</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">NN</font></span>)</span>
 
<b>by </b><i><a class="ref" href="radix_1.html#T3" target="_self">Th3</a>, <a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>;<br/>
<b>set </b><font color="Maroon">I</font> = <font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1;<br/>
<a NAME="E14:35_2_1"/><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/>
<a NAME="E15:35_2_1"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K9" target="_self">DigitDC</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="radix_1.html#E1:35"><i><font color="Green">E29</font></i></a>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>;<br/>
<a NAME="E16:35_2_1"/><i><font color="Green">E100</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E17:35_2_1"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E18:35_2_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<i><font color="Green">E103</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span> = 
 <a href="radix_1.html#K6" target="_self">SubDigit</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
;<br/>
<a NAME="E20:35_2_1"/>
<span class="p1">(<span class="default"><a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">u</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><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" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:35_2_1_2"/><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="mesfunc1.html#K2">INT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">z0</font> =  <a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">u</font>,<font color="Maroon">k</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">z1</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><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" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a>  <b>by </b><i>, <a class="ref" href="finseq_1.html#T95">FINSEQ_1:95</a></i>;<br/>
<b>reconsider </b><font color="Maroon">DD</font> =  <a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <span class="p1">(<span class="default"><a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">u</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">z1</font> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a>  ;<br/>
<i><font color="Green">E107</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">u</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><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" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><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" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/>
<a NAME="E6:35_2_1_2"/><i><font color="Green">E108</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">DD</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E7:35_2_1_2"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> holds <br/><font color="Maroon">z</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">DD</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:35_2_1_2_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> )
 ;<br/>

<a NAME="E2:35_2_1_2_2"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_2_1_2_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">u</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="radix_1.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_2.html#T8">FINSEQ_2:8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:35_2_1_2_2_1_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">u</font>
 ;<br/><div class="add"><a NAME="E2:35_2_1_2_2_1_1"/><i><font color="Green">E111</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">i</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_2_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:35_2_1_2_2_1_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">u</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E2:35_2_1_2_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:35_2_1_2_2_1_1_1"/><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E5:35_2_1_2_2_1_1_1"/>
 <a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E6:35_2_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="prepower.html#T12">PREPOWER:12</a></i>;<br/>
<b>hence </b><a NAME="E7:35_2_1_2_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E3:35_2_1_2_2_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">u</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/><a NAME="E4:35_2_1_2_2_1_1"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">u</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><i><font color="Green">E114</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">u</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">z1</font></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">u</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">u</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="radix_1.html#K6" target="_self">SubDigit</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">u</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">k</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K9" target="_self">DigitDC</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/><a NAME="E6:35_2_1_2_2_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">DD</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/><a NAME="E7:35_2_1_2_2_1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">DD</font>
 <b>by </b><i><a class="ref" href="radix_1.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><i><font color="Green">E116</font></i>: <font color="Maroon">DD</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="radix_1.html#K6" target="_self">SubDigit</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="radix_1.html#T6" target="_self">Th6</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K9" target="_self">DigitDC</a> <font color="Maroon">p</font>,<font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="radix_1.html#T6" target="_self">Th6</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>

;<br/><a NAME="E9:35_2_1_2_2_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">DD</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">DD</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E10:35_2_1_2_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">u</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">z1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="radix_1.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E11:35_2_1_2_2_1_1"/>
<font color="Maroon">z</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">DD</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="radix_1.html#E2:35"><i><font color="Green">E47</font></i></a>, <a class="txt" href="radix_1.html#E1:35_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="radix_1.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:35_2_1_2_2_1_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><font color="Maroon">z</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<font color="Maroon">z</font> <a href="wsierp_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">u</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">z1</font></span>)</span> <a href="wsierp_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">u</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
<br/>.= 
<font color="Maroon">DD</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E8:35_2_1_2"/>
<span class="p1">(<span class="default"><a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">u</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><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" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K7" target="_self">DigitSD</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_5.html#T14">FINSEQ_5:14</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E21:35_2_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K8" target="_self">SDDec</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="radix_1.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:35_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:35"/>
for <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">u</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">u</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S9">NAT_1:sch 9</a>(, );<br/></i>
<b>hence </b><a NAME="E4:35"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="radix_1.html#R1" target="_self">is_represented_by</a> <font color="Olive">n</font>,<font color="Maroon">k</font> holds <br/><font color="Olive">m</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K8" target="_self">SDDec</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" target="_self">DecSD</a> <font color="Olive">m</font>,<font color="Olive">n</font>,<font color="Maroon">k</font></span>)</span>
 ;<br/>


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