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

<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="radix_3.html#K6">SDSub_Add_Data</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="radix_3.html#K5">SDSub_Add_Carry</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E1:11_1_1"/>
 <a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> <a href="radix_3.html#K4">-SD_Sub</a> 
 
<b>by </b><i><a class="txt" href="radix_4.html#E1:11"><i><font color="Green">E11</font></i></a>, <a class="ref" href="radix_3.html#T19">RADIX_3:19</a></i>;<br/>
<a NAME="E2:11_1_1"/><b>then </b><i><font color="Green">E12</font></i>: 
 <a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="radix_3.html#K4">-SD_Sub</a> 
 
;<br/>
<a NAME="E3:11_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="radix_3.html#K4">-SD_Sub</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="radix_1.html#K3">-SD</a> 
 
<b>by </b><i><a class="txt" href="radix_4.html#E1:11"><i><font color="Green">E11</font></i></a>, <a class="ref" href="radix_3.html#T5">RADIX_3:5</a></i>;<br/>
<a NAME="E4:11_1_1"/>
 <a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> <a href="radix_3.html#K4">-SD_Sub</a> 
 
<b>by </b><i><a class="txt" href="radix_4.html#E1:11"><i><font color="Green">E11</font></i></a>, <a class="ref" href="radix_3.html#T19">RADIX_3:19</a></i>;<br/>
<a NAME="E5:11_1_1"/><b>then </b>
 <a href="radix_3.html#K6">SDSub_Add_Data</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="radix_3.html#K3">-SD_Sub_S</a> 
 
<b>by </b><i><a class="txt" href="radix_4.html#E1:11"><i><font color="Green">E11</font></i></a>, <a class="txt" href="radix_4.html#E2:11_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="radix_4.html#E3:11_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="radix_3.html#T20">RADIX_3:20</a></i>;<br/>
<a NAME="E6:11_1_1"/><b>then </b><i><font color="Green">E14</font></i>: 
(  <a href="radix_3.html#K6">SDSub_Add_Data</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> &amp;  <a href="radix_3.html#K6">SDSub_Add_Data</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 )
 
<b>by </b><i><a class="txt" href="radix_4.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E7:11_1_1"/><i><font color="Green">E15</font></i>: 
(  <a href="xcmplx_0.html#K4">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="radix_3.html#K5">SDSub_Add_Carry</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="radix_3.html#K5">SDSub_Add_Carry</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="radix_4.html#E1:11"><i><font color="Green">E11</font></i></a>, <a class="ref" href="radix_3.html#T13">RADIX_3:13</a></i>;<br/>
<a NAME="E8:11_1_1"/><b>then </b><i><font color="Green">E16</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> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></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#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="radix_3.html#K6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="radix_4.html#E6:11_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E9:11_1_1"/><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><a href="radix_3.html#K6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <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">Radix</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="txt" href="radix_4.html#E6:11_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="radix_4.html#E7:11_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E10:11_1_1"/><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><a href="radix_3.html#K6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="wsierp_1.html#K6">INT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<a NAME="E11:11_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="radix_3.html#K4">-SD_Sub</a>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="wsierp_1.html#K6">INT</a>  : ( <font color="Olive">b<sub>1</sub></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> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="radix_1.html#K1">Radix</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="radix_3.html#D2">RADIX_3:def 2</a></i>;<br/>
<a NAME="E12:11_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="radix_3.html#K6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="radix_3.html#K4">-SD_Sub</a> 
 
<b>by </b><i><a class="txt" href="radix_4.html#E8:11_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="radix_4.html#E9:11_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="radix_4.html#E10:11_1_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:11_1_1"/>
<span class="p1">(<span class="default"><a href="radix_3.html#K6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_3.html#K7">DigA_SDSub</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> <a href="radix_3.html#K4">-SD_Sub</a> 
 ;<br/>


</div>
