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

<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>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a> </span>)</span>;<br/>







<b>assume </b><a NAME="E1:30"/><i><font color="Green">E30</font></i>: 
( 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/>

<b>set </b><font color="Maroon">SDC</font> =  <a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font>;<br/>
<b>set </b><font color="Maroon">SDD</font> = <span class="p1">(<span class="default"><a href="radix_3.html#K6" target="_self">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">a</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">a</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>;<br/>
<a NAME="E2:30"/><i><font color="Green">E35</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:30"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="finseq_2.html#T8">FINSEQ_2:8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:30_1_1_1"/>
<span class="p1">(<span class="default"><a href="radix_3.html#K6" target="_self">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">a</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">a</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="radix_3.html#K4" target="_self">-SD_Sub</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1_1_1"/><i><font color="Green">E41</font></i>: 
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">a</font>,<font color="Maroon">i</font> 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="ref" href="radix_1.html#T19">RADIX_1:19</a></i>;<br/>
<a NAME="E2:30_1_1_1_1"/><i><font color="Green">E47</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="radix_3.html#E1:30"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_1_1_1_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:30_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_2.html#T10">NAT_2:10</a></i>;<br/><a NAME="E3:30_1_1_1_1_1_1_1"/><b>then </b>
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>;<br/><a NAME="E4:30_1_1_1_1_1_1_1"/><b>then </b>
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <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="ref" href="radix_1.html#T16">RADIX_1:16</a></i>;<br/><b>hence </b><a NAME="E5:30_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="radix_3.html#K6" target="_self">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">a</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">a</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="radix_3.html#K4" target="_self">-SD_Sub</a> 
 <b>by </b><i>, <a class="txt" href="radix_3.html#E1"><i><font color="Green">Lemma29</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_1_1_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:30_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E3:30_1_1_1_1_1_1_2"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E4:30_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="txt" href="radix_3.html#E2"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:30_1_1_1_1_1_1_2"/><b>then </b>
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> 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="ref" href="radix_1.html#T19">RADIX_1:19</a></i>;<br/><b>hence </b><a NAME="E6:30_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="radix_3.html#K6" target="_self">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">a</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">a</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="radix_3.html#K4" target="_self">-SD_Sub</a> 
 <b>by </b><i>, <a class="txt" href="radix_3.html#E1"><i><font color="Green">Lemma29</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:30_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="radix_3.html#K6" target="_self">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">a</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">a</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="radix_3.html#K4" target="_self">-SD_Sub</a> 
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:30_1_1_1"/>
 <a href="radix_3.html#K8" target="_self">SD2SDSubDigit</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="radix_3.html#K4" target="_self">-SD_Sub</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:30_1_1_2"/>
 <a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="radix_3.html#K4" target="_self">-SD_Sub</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1_2_1"/><i><font color="Green">E50</font></i>: 
 <a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="radix_3.html#K3" target="_self">-SD_Sub_S</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:30_1_1_2_1"/>
<font color="Maroon">k</font> <a href="radix_3.html#K3" target="_self">-SD_Sub_S</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> <a href="radix_3.html#K4" target="_self">-SD_Sub</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:30_1_1_2_1"/>
 <a href="radix_3.html#K5" target="_self">SDSub_Add_Carry</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="radix_3.html#K4" target="_self">-SD_Sub</a> 
 <b>by </b><i><a class="txt" href="radix_3.html#E1"><i><font color="Green">Lemma29</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:30_1_1_2"/>
 <a href="radix_3.html#K8" target="_self">SD2SDSubDigit</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="radix_3.html#K4" target="_self">-SD_Sub</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:30"/>
 <a href="radix_3.html#K8" target="_self">SD2SDSubDigit</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">k</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">k</font> <a href="radix_3.html#K4" target="_self">-SD_Sub</a> 
 ;<br/>


</div>
