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

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><b>let </b><font color="Maroon" title="c3">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:5"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3
 ;<br/>

<a NAME="E2:5"/><b>then </b><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<a NAME="E3:5"/><b>then </b>
<font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:5"/><b>then </b>
<font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E5:5"/><b>then </b><i><font color="Green" title="E8">A3</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="c3">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 
<b>by </b><i><a class="txt" href="radix_4.html#E2:5"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="radix_4.html#T1" target="_self" title="RADIX_4:th.1">Th1</a></i>;<br/>
<a NAME="E6:5"/><b>then </b><i><font color="Green" title="E9">A4</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="c3">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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">CC</font> = <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">k</font></span>)</span>;<br/>
<a NAME="E7:5"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<b>by </b><i><a class="txt" href="radix_4.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:5"/><b>then </b><i><font color="Green" title="E11">A6</font></i>: 
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">k</font> &amp;  <a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="radix_3.html#T13" title="RADIX_3:th.13">RADIX_3:13</a></i>;<br/>
<a NAME="E9:5"/><i><font color="Green" title="E12">A7</font></i>: 
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">k</font> &amp;  <a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="radix_4.html#E7:5"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="radix_3.html#T13" title="RADIX_3:th.13">RADIX_3:13</a></i>;<br/>
<a NAME="E10:5"/><b>then </b>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</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> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="radix_4.html#E8:5"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E11:5"/><b>then </b><i><font color="Green" title="E13">A8</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="c3">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> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="radix_4.html#E6:5"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:5"/>
<span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="radix_4.html#E8:5"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="radix_4.html#E9:5"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E13:5"/><b>then </b>
<span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">k</font></span>)</span> <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="c3">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="radix_4.html#E5:5"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E14:5"/>
 <a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">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="txt" href="radix_4.html#E11:5"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="radix_3.html#D3" title="RADIX_3:def.3">RADIX_3:def 3</a></i>;<br/>


</div>
