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

<b>let </b><font color="Maroon" title="c1">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i1</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Olive" title="b1">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a>  holds <br/>(  <a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <font color="Olive" title="b1">i1</font>,<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> &amp;  <a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <font color="Olive" title="b1">i1</font>,<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )</span><br/><b>let </b><font color="Maroon" title="c2">i1</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c2">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a>  implies (  <a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> &amp;  <a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 ) )</span><br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E19">A1</font></i>: 
( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c2">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> &amp;  <a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )</span><br/>

<a NAME="E2:24"/><b>then </b><i><font color="Green" title="E20">A2</font></i>: 
( <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i1</font> &amp; <font color="Maroon" title="c2">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<b>by </b><i><a class="ref" href="radix_1.html#T15" title="RADIX_1:th.15">RADIX_1:15</a></i>;<br/>
<a NAME="E3:24"/><i><font color="Green" title="E21">A3</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="txt" href="radix_3.html#E1:24"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1"/>
( <font color="Maroon" title="c2">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> or (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i1</font> &amp; <font color="Maroon" title="c2">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) or  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1"><b>case </b></a><a NAME="E1:24_1_1_1"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c2">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> &amp;  <a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )</span><br/><div class="add"><a NAME="E2:24_1_1_1"/><b>then </b><i><font color="Green" title="E23">A5</font></i>: 
 <a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 <b>by </b><i><a class="ref" href="radix_3.html#D3" target="_self" title="RADIX_3:def.3">Def3</a></i>;<br/><a NAME="E3:24_1_1_1"/>
<font color="Maroon" title="c2">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="radix_3.html#E1:24_1_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><a NAME="E4:24_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E5:24_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E6:24_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><b>hence </b><a NAME="E7:24_1_1_1"/>
(  <a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> &amp;  <a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 <b>by </b><i><a class="txt" href="radix_3.html#E2:24"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="radix_3.html#E3:24"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="radix_3.html#E2:24_1_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="radix_3.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2"><b>case </b></a><a NAME="E1:24_1_1_2"/><i><font color="Green" title="E22">A6</font></i>: 
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i1</font> &amp; <font color="Maroon" title="c2">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> &amp;  <a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )</span><br/><div class="add"><a NAME="E2:24_1_1_2"/><b>then </b><i><font color="Green" title="E23">A7</font></i>: 
 <a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="radix_3.html#D3" target="_self" title="RADIX_3:def.3">Def3</a></i>;<br/><a NAME="E3:24_1_1_2"/>
<font color="Maroon" title="c2">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="radix_3.html#E1:24_1_1_2"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><b>hence </b><a NAME="E4:24_1_1_2"/>
(  <a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> &amp;  <a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 <b>by </b><i><a class="txt" href="radix_3.html#E1:24_1_1_2"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="radix_3.html#E2:24_1_1_2"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_3"><b>case </b></a><a NAME="E1:24_1_1_3"/><i><font color="Green" title="E22">A8</font></i>: 
 <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> &amp;  <a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )</span><br/><div class="add"><a NAME="E2:24_1_1_3"/><b>then </b><i><font color="Green" title="E23">A9</font></i>: 
 <a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="radix_3.html#D3" target="_self" title="RADIX_3:def.3">Def3</a></i>;<br/><a NAME="E3:24_1_1_3"/><i><font color="Green" title="E24">A10</font></i>: 
<span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="radix_3.html#E1:24_1_1_3"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E4:24_1_1_3"/><i><font color="Green" title="E25">A11</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E5:24_1_1_3"/>
<font color="Maroon" title="c2">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="radix_3.html#E1:24"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="radix_1.html#T15" title="RADIX_1:th.15">RADIX_1:15</a></i>;<br/><a NAME="E6:24_1_1_3"/><b>then </b>
<font color="Maroon" title="c2">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><b>hence </b><a NAME="E7:24_1_1_3"/>
(  <a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> &amp;  <a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 <b>by </b><i><a class="txt" href="radix_3.html#E1:24"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="radix_3.html#E2:24_1_1_3"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="radix_3.html#E3:24_1_1_3"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="radix_3.html#E4:24_1_1_3"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="radix_3.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:24"/>
(  <a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> &amp;  <a href="radix_3.html#K6" title="RADIX_3:func.6">SDSub_Add_Data</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
