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

<a NAME="E1:29_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  : ( <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <font color="Olive">w</font> <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">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="radix_1.html#D2">RADIX_1:def 2</a></i>;<br/>
<a NAME="E2:29_1_1"/><i><font color="Green">E26</font></i>: 
<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">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:29_1_1_1"/>
 <a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2
 
<b>by </b><i>, <a class="ref" href="radix_4.html#T1">RADIX_4:1</a></i>;<br/>
<a NAME="E2:29_1_1_1"/><b>then </b>
 <a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:29_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E4:29_1_1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="real_2.html#T169">REAL_2:169</a></i>;<br/>
<a NAME="E5:29_1_1_1"/>
<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">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:29_1_1_1"/>
<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">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a> 
 <b>by </b><i><a class="txt" href="radix_6.html#E1:29_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="radix_6.html#E4:29_1_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:29_1_1"/>
1 <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:29_1_1_2"/><i><font color="Green">E29</font></i>: 
 <a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2
 
<b>by </b><i>, <a class="ref" href="radix_4.html#T1">RADIX_4:1</a></i>;<br/>
<a NAME="E2:29_1_1_2"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:29_1_1_2"/>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> 2
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E4:29_1_1_2"/><b>then </b>
<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">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:29_1_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<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">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:29_1_1_2"/>
1 is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:29_1_1_2"/>
1 <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a> 
 <b>by </b><i><a class="txt" href="radix_6.html#E1:29_1_1"><i><font color="Green">E25</font></i></a>, , <a class="txt" href="radix_6.html#E1:29_1_1_2"><i><font color="Green">E29</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:29_1_1"/>
( ( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font> implies 0 is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a>  ) &amp; ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> implies 1 is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a>  ) &amp; ( not <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font> &amp; not <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> implies <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">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a>  ) )
 <b>by </b><i><a class="txt" href="radix_6.html#E2:29_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="radix_1.html#T16">RADIX_1:16</a></i>;<br/>


</div>
