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

<b>set </b><font color="Maroon">T</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">i</font>;<br/>
<b>set </b><font color="Maroon">S</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> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>;<br/>
<a NAME="E1:27_1_1"/><i><font color="Green">E29</font></i>: 
 <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="E2:27_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E3:27_1_1"/>
<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="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="prepower.html#T19">PREPOWER:19</a></i>;<br/>
<a NAME="E4:27_1_1"/><b>then </b><i><font color="Green">E37</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">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E5:27_1_1"/>
<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">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="prepower.html#T19">PREPOWER:19</a></i>;<br/>
<a NAME="E6:27_1_1"/><b>then </b><i><font color="Green">E38</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> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E7:27_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
0 <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> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T80">NAT_1:80</a></i>;<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><span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></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">i</font></span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">k</font> <a href="radix_1.html#K3" target="_self">-SD</a> ;<br/>
<a NAME="E8:27_1_1"/>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="radix_1.html#K3" target="_self">-SD</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E9:27_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>;<br/>
<a NAME="E10:27_1_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:27_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">xx</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">i</font></span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">k</font> <a href="radix_1.html#K3" target="_self">-SD</a> 
 ;<br/>

<b>set </b><font color="Maroon">X</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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">i</font></span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>;<br/>
<a NAME="E2:27_1_1_1"/>
<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="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<b>then reconsider </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="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><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> = 
<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="power.html#K3">to_power</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="power.html#T46">POWER:46</a></i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="power.html#T29">POWER:29</a></i>
;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="nat_1.html#K1">+</a> 1</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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> 1

<br/>.= 
0
<b>by </b><i><a class="ref" href="nat_1.html#T63">NAT_1:63</a></i>
;<br/><a NAME="E4:27_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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">i</font></span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="nat_1.html#T80">NAT_1:80</a></i>;<br/><b>hence </b><a NAME="E5:27_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">xx</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:27_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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">i</font></span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</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="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, </i>;<br/><a NAME="E3:27_1_1_1_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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">i</font></span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:27_1_1_1_1_2"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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">i</font></span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</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/><a NAME="E5:27_1_1_1_1_2"/>
 <a href="xcmplx_0.html#K4">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E6:27_1_1_1_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="real_2.html#T109">REAL_2:109</a></i>;<br/><a NAME="E7:27_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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">i</font></span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="radix_1.html#K3" target="_self">-SD</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E8:27_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">xx</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E11:27_1_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E12:27_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" 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="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> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">k</font> <a href="radix_1.html#K3" target="_self">-SD</a> 
 ;<br/>


</div>
