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

<b>let </b><font color="Maroon">s1</font> be   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">n</font>,<a href="mesfunc1.html#K2">INT</a> , <font color="Maroon">s2</font> be   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">n</font>,<a href="mesfunc1.html#K2">INT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:22_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="radix_2.html#K6" target="_self">Table1</a> <font color="Maroon">q</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font>,<font color="Maroon">n</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">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> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 holds <br/> ex <font color="Olive">I1</font>, <font color="Olive">I2</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">I1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">I2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">I2</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">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">I1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_2.html#K6" target="_self">Table1</a> <font color="Maroon">q</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Olive">i</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">f</font> ) ) )
 <b>and </b><br/><a NAME="E2:22_1_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="radix_2.html#K6" target="_self">Table1</a> <font color="Maroon">q</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font>,<font color="Maroon">n</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">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> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 holds <br/> ex <font color="Olive">I1</font>, <font color="Olive">I2</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">I1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">I2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">I2</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">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">I1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_2.html#K6" target="_self">Table1</a> <font color="Maroon">q</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Olive">i</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">f</font> ) ) )
 ;<br/>

<a NAME="E3:22_1_2"/><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E4:22_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s2</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 implies <font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E5:22_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E6:22_1_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">kk</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:22_1_2_1"/><i><font color="Green">E53</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 implies <font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font> )
 ;<br/>

<b>assume </b><a NAME="E2:22_1_2_1"/><i><font color="Green">E54</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 )
 ;<br/>

<a NAME="E3:22_1_2_1"/><i><font color="Green">E55</font></i>: 
( 0 <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> or ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">kk</font> &amp; 0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 ) )
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_2_1_1"/>
( 0 <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> or 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> )
 </span><b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_2_1_1_1"/>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_1_2_1_1_1"/>
<font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_2_1_1_2"/><i><font color="Green">E56</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font>
 ;<br/><div class="add"><a NAME="E2:22_1_2_1_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:22_1_2_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 )
 <b>by </b><i>, <a class="ref" href="radix_2.html#T3" target="_self">Th3</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:22_1_2_1_1_2"/><b>then </b><i><font color="Green">E65</font></i>: 
 ex <font color="Olive">I1</font>, <font color="Olive">I2</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">I1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font> &amp; <font color="Olive">I2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">I2</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">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">I1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_2.html#K6" target="_self">Table1</a> <font color="Maroon">q</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">kk</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">f</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">i1</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">i2</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E6:22_1_2_1_1_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">i2</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">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_2.html#K6" target="_self">Table1</a> <font color="Maroon">q</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">kk</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">f</font> )
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E7:22_1_2_1_1_2"/>
<font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">kk</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="radix_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="radix_2.html#T3" target="_self">Th3</a>, <a class="ref" href="radix_2.html#T4" target="_self">Th4</a>, <a class="txt" href="radix_2.html#E1:22"><i><font color="Green">E29</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E7:22_1_2"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">kk</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">kk</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E8:22_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:22_1_2_2"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">U1</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">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="hs2(this)" href="javascript:()" title="22_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_2_2_1_1"/>
( <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">U1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_2_2_1_1_1"/>
<font color="Maroon">U1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_2_2_1_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><b>hence </b><a NAME="E3:22_1_2_2_1_1_1"/>
<font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_2_2_1_1_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">U1</font>
 ;<br/><div class="add"><a NAME="E2:22_1_2_2_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">U1</font> &amp; <font color="Maroon">U1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">U1</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>then consider </b><font color="Maroon">i1</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">i2</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E4:22_1_2_2_1_1_2"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">U1</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">U1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>and </b><br/><a NAME="E5:22_1_2_2_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">i2</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">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_2.html#K6" target="_self">Table1</a> <font color="Maroon">q</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">U1</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">j1</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">j2</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E7:22_1_2_2_1_1_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">U1</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">U1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>and </b><br/><a NAME="E8:22_1_2_2_1_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">j2</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">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_2.html#K6" target="_self">Table1</a> <font color="Maroon">q</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">U1</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E9:22_1_2_2_1_1_2"/>
<font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="radix_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="radix_2.html#T3" target="_self">Th3</a>, <a class="ref" href="radix_2.html#T4" target="_self">Th4</a>, , <a class="txt" href="radix_2.html#E1:22"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:22_1_2_2"/>
<font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:22_1_2"/>
for <font color="Olive">kk</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font> holds <br/><font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">kk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">kk</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">kk</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:22_1_2_3"/><i><font color="Green">E76</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_2_3_1_1"/>
( <font color="Maroon">kk</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font> or <font color="Maroon">kk</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font> )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_2_3_1_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">kk</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font>
 ;<br/><div class="add"><a NAME="E2:22_1_2_3_1_1_1"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon">U1</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">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/><a NAME="E4:22_1_2_3_1_1_1"/>
<font color="Maroon">U1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font>
 ;<br/><a NAME="E5:22_1_2_3_1_1_1"/><b>then </b>
<font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">U1</font>
 <b>by </b><i><a class="ref" href="radix_2.html#T3" target="_self">Th3</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E6:22_1_2_3_1_1_1"/>
<font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font>
 <b>by </b><i>, <a class="ref" href="radix_2.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_2_3_1_1_2"/>
<font color="Maroon">kk</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_1_2_3_1_1_2"/>
<font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:22_1_2_3"/>
<font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:22_1_2"/>
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div>
