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

<b>let </b><font color="Maroon">i1</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">i2</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><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>assume </b><a NAME="E1:32"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="radix_1.html#K3" target="_self">-SD</a>  &amp; <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="radix_1.html#K3" target="_self">-SD</a>  )
 ;<br/>

<a NAME="E2:32"/><b>then </b><i><font color="Green">E30</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" target="_self">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> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&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="xcmplx_0.html#K6">-</a> 1 &amp; <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 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&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="xcmplx_0.html#K6">-</a> 1 )
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">z</font> = <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font>;<br/>
<a NAME="E3:32"/><i><font color="Green">E37</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<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</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&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> <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> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1"/>
( <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> 2 or (  <a href="xcmplx_0.html#K4">-</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 ) or <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1"><b>suppose </b></a><a NAME="E1:32_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> 2
 ;<br/><div class="add"><a NAME="E2:32_1_1"/><b>then </b>
 <a href="radix_1.html#K11" target="_self">SD_Add_Carry</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> 1
 <b>by </b><i/>;<br/><a NAME="E3:32_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="radix_1.html#K12" target="_self">SD_Add_Data</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font></span>)</span>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span>
 ;<br/><a NAME="E4:32_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E5:32_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<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> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</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="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font>
 ;<br/><a NAME="E6:32_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font></span>)</span> <a href="xcmplx_0.html#K2">+</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="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> <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#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E7:32_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" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="radix_1.html#K12" target="_self">SD_Add_Data</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font></span>)</span>,<font color="Maroon">k</font> &amp;  <a href="radix_1.html#K12" target="_self">SD_Add_Data</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font></span>)</span>,<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&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="xcmplx_0.html#K6">-</a> 2 )
 <b>by </b><i><a class="ref" href="radix_1.html#T2" target="_self">Th2</a>, <a class="txt" href="radix_1.html#E1:32"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2"><b>suppose </b></a><a NAME="E1:32_1_2"/><i><font color="Green">E49</font></i>: 
(  <a href="xcmplx_0.html#K4">-</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 )
 ;<br/><div class="add"><a NAME="E2:32_1_2"/><b>then </b>
 <a href="radix_1.html#K11" target="_self">SD_Add_Carry</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E3:32_1_2"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="radix_1.html#K12" target="_self">SD_Add_Data</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font></span>)</span>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font>
 ;<br/><a NAME="E4:32_1_2"/><i><font color="Green">E81</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> 2 <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i>, </i>;<br/><a NAME="E5:32_1_2"/><b>then </b><i><font color="Green">E82</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="xcmplx_0.html#K6">-</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:32_1_2"/>
 <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> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="radix_1.html#E1:32"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E7:32_1_2"/><b>then </b>
 <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> <a href="xxreal_0.html#R1">&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> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span>
 ;<br/><a NAME="E8:32_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" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xcmplx_0.html#K4">-</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><b>hence </b><a NAME="E9:32_1_2"/>
( <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> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="radix_1.html#K12" target="_self">SD_Add_Data</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font></span>)</span>,<font color="Maroon">k</font> &amp;  <a href="radix_1.html#K12" target="_self">SD_Add_Data</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font></span>)</span>,<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&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="xcmplx_0.html#K6">-</a> 2 )
 <b>by </b><i>, <a class="ref" href="radix_1.html#T2" target="_self">Th2</a>, <a class="txt" href="radix_1.html#E2:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_3"><b>suppose </b></a><a NAME="E1:32_1_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2
 ;<br/><div class="add"><a NAME="E2:32_1_3"/><b>then </b>
 <a href="radix_1.html#K11" target="_self">SD_Add_Carry</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/><a NAME="E3:32_1_3"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="radix_1.html#K12" target="_self">SD_Add_Data</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font></span>)</span>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" target="_self">Radix</a> <font color="Maroon">k</font></span>)</span>
 ;<br/><a NAME="E4:32_1_3"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</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="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</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/>;<br/><a NAME="E5:32_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font></span>)</span> <a href="xcmplx_0.html#K6">-</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="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="xcmplx_0.html#K6">-</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#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E6:32_1_3"/>
( <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> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="radix_1.html#K12" target="_self">SD_Add_Data</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font></span>)</span>,<font color="Maroon">k</font> &amp;  <a href="radix_1.html#K12" target="_self">SD_Add_Data</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font></span>)</span>,<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&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="xcmplx_0.html#K6">-</a> 2 )
 <b>by </b><i><a class="ref" href="radix_1.html#T2" target="_self">Th2</a>, <a class="txt" href="radix_1.html#E1:32"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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