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

<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> , <font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <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="E1:29"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 ;<br/>

<a NAME="E2:29"/><b>then </b><i><font color="Green">E21</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:29"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:29"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E5:29"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<b>then </b><i><font color="Green">E43</font></i>:  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="radix_5.html#K5" target="_self">FminDigit</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
0
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, , <a class="ref" href="radix_5.html#T1" target="_self">Th1</a></i>
;<br/>
<i><font color="Green">E44</font></i>:  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="radix_5.html#K7" target="_self">FmaxDigit</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="radix_5.html#T3" target="_self">Th3</a>, <a class="txt" href="radix_5.html#E2:29"><i><font color="Green">E21</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, , <a class="txt" href="radix_5.html#E1:29"><i><font color="Green">E20</font></i></a></i>
;<br/>
<i><font color="Green">E47</font></i>: <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <span class="p2">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="radix_1.html#K14">'+'</a> <span class="p3">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span></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#K11">SD_Add_Carry</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p5">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p5">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, , <a class="ref" href="radix_2.html#T11">RADIX_2:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="radix_1.html#K14">'+'</a> <span class="p3">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="radix_1.html#T21">RADIX_1:21</a></i>
;<br/>
<a NAME="E9:29"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">m</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>
 
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/>
<a NAME="E10:29"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">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> holds <br/> <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="radix_1.html#K14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_4"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:29_4"/><i><font color="Green">E51</font></i>: 
<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>then </b><i><font color="Green">E52</font></i>:  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="radix_1.html#K14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font> = 
 <a href="radix_1.html#K13">Add</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="radix_1.html#D14">RADIX_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K12">SD_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p4">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p4">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font></span>)</span></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_1.html#K11">SD_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p4">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p4">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, , <a class="ref" href="radix_1.html#D13">RADIX_1:def 13</a></i>
;<br/>
<a NAME="E3:29_4"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:29_4"/><i><font color="Green">E54</font></i>: 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_5.html#K5" target="_self">FminDigit</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>,<font color="Maroon">i</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:29_4"/><i><font color="Green">E55</font></i>: 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_5.html#K7" target="_self">FmaxDigit</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>,<font color="Maroon">i</font>
 
<b>by </b><i>, <a class="txt" href="radix_5.html#E2:29"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E6:29_4"/><i><font color="Green">E56</font></i>: 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_5.html#K5" target="_self">FminDigit</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">i</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:29_4"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<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="29_4_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_4_2_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_2_1_1"><b>suppose </b></a><a NAME="E1:29_4_2_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_4_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_2_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="E1:29_4_2_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_4_2_1_1_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green">E62</font></i>:  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> = 
 <a href="radix_5.html#K5" target="_self">FminDigit</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="radix_5.html#T7" target="_self">Th7</a>, <a class="ref" href="radix_5.html#T1" target="_self">Th1</a></i>
;<br/><i><font color="Green">E63</font></i>:  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> = 
 <a href="radix_5.html#K7" target="_self">FmaxDigit</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="txt" href="radix_5.html#E13"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="radix_5.html#E2:29"><i><font color="Green">E21</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="radix_5.html#T7" target="_self">Th7</a>, <a class="txt" href="radix_5.html#E1:29"><i><font color="Green">E20</font></i></a></i>
;<br/><i><font color="Green">E64</font></i>:  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>
<b>by </b><i><a class="txt" href="radix_5.html#E13"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
 <a href="radix_5.html#K5" target="_self">FminDigit</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>,<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="radix_5.html#T4" target="_self">Th4</a>, </i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="radix_5.html#T1" target="_self">Th1</a></i>
;<br/><i><font color="Green">E65</font></i>:  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>
<b>by </b><i><a class="txt" href="radix_5.html#E13"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
 <a href="radix_5.html#K7" target="_self">FmaxDigit</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>,<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="radix_5.html#T4" target="_self">Th4</a>, <a class="txt" href="radix_5.html#E2:29"><i><font color="Green">E21</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="txt" href="radix_5.html#E1:29"><i><font color="Green">E20</font></i></a></i>
;<br/><i><font color="Green">E66</font></i>:  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> = 
 <a href="radix_5.html#K5" target="_self">FminDigit</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="txt" href="radix_5.html#E13"><i><font color="Green">Lemma40</font></i></a>, </i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="radix_5.html#T1" target="_self">Th1</a></i>
;<br/> <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="radix_1.html#K14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font> = 
0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K11">SD_Add_Carry</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="radix_5.html#T5" target="_self">Th5</a>, <a class="ref" href="radix_5.html#T12" target="_self">Th12</a>, , <a class="txt" href="radix_5.html#E6:29"><i><font color="Green">E43</font></i></a>, <a class="txt" href="radix_5.html#E7:29"><i><font color="Green">E44</font></i></a>, <a class="ref" href="radix_1.html#T22">RADIX_1:22</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E8:29_4_2_1_1_1_1_1"/>
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="radix_1.html#K14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_4_2_1_1_1_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:29_4_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="radix_5.html#E12"><i><font color="Green">Lemma39</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:29_4_2_1_1_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="radix_5.html#E12"><i><font color="Green">Lemma39</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:29_4_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="radix_5.html#T2" target="_self">Th2</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:29_4_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<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> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E6:29_4_2_1_1_1_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="radix_5.html#T12" target="_self">Th12</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:29_4_2_1_1_1_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="radix_5.html#E6:29"><i><font color="Green">E43</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E8:29_4_2_1_1_1_1_2"/><i><font color="Green">E72</font></i>: 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="radix_5.html#T7" target="_self">Th7</a>, <a class="ref" href="radix_5.html#T8" target="_self">Th8</a>, <a class="txt" href="radix_5.html#E12"><i><font color="Green">Lemma39</font></i></a>, <a class="ref" href="radix_5.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E9:29_4_2_1_1_1_1_2"/><i><font color="Green">E73</font></i>: 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="radix_5.html#T7" target="_self">Th7</a>, <a class="txt" href="radix_5.html#E5:29"><i><font color="Green">E36</font></i></a>, <a class="txt" href="radix_5.html#E12"><i><font color="Green">Lemma39</font></i></a>, <a class="txt" href="radix_5.html#E1:29"><i><font color="Green">E20</font></i></a></i>;<br/><i><font color="Green">E74</font></i>:  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
 <a href="radix_5.html#K5" target="_self">FminDigit</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="txt" href="radix_5.html#E7:29"><i><font color="Green">E44</font></i></a>, <a class="ref" href="radix_5.html#T1" target="_self">Th1</a></i>
;<br/><i><font color="Green">E75</font></i>:  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
 <a href="radix_5.html#K7" target="_self">FmaxDigit</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="txt" href="radix_5.html#E2:29"><i><font color="Green">E21</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="txt" href="radix_5.html#E7:29"><i><font color="Green">E44</font></i></a>, <a class="txt" href="radix_5.html#E1:29"><i><font color="Green">E20</font></i></a></i>
;<br/><a NAME="E12:29_4_2_1_1_1_1_2"/>
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="radix_5.html#T10" target="_self">Th10</a>, <a class="txt" href="radix_5.html#E13"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="radix_5.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E13:29_4_2_1_1_1_1_2"/>
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="radix_1.html#K14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="radix_5.html#T5" target="_self">Th5</a>, , , <a class="txt" href="radix_5.html#E8:29"><i><font color="Green">E47</font></i></a>, <a class="txt" href="radix_5.html#E9:29"><i><font color="Green">E48</font></i></a>, <a class="ref" href="radix_1.html#T21">RADIX_1:21</a>, <a class="ref" href="radix_1.html#T22">RADIX_1:22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:29_4_2_1_1"/>
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="radix_1.html#K14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_2_1_2"><b>suppose </b></a><a NAME="E1:29_4_2_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:29_4_2_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_4_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_4_2_1_2_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="ref" href="radix_5.html#T11" target="_self">Th11</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:29_4_2_1_2_1_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:29_4_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="radix_5.html#E12"><i><font color="Green">Lemma39</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:29_4_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1 <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>
 <b>by </b><i><a class="ref" href="radix_5.html#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E4:29_4_2_1_2_1_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="radix_5.html#T12" target="_self">Th12</a>, <a class="ref" href="jordan5b.html#T1">JORDAN5B:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_4_2_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_2_1_2_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="E1:29_4_2_1_2_1_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 </span><b>by </b><i><a class="txt" href="radix_5.html#E12"><i><font color="Green">Lemma39</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_2_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_4_2_1_2_1_1_1_1_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:29_4_2_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="txt" href="radix_5.html#E5:29"><i><font color="Green">E36</font></i></a>, <a class="txt" href="radix_5.html#E1:29"><i><font color="Green">E20</font></i></a></i>;<br/><i><font color="Green">E90</font></i>:  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
 <a href="radix_5.html#K5" target="_self">FminDigit</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="txt" href="radix_5.html#E7:29"><i><font color="Green">E44</font></i></a>, </i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="txt" href="radix_5.html#E6:29"><i><font color="Green">E43</font></i></a>, <a class="ref" href="radix_5.html#T1" target="_self">Th1</a></i>
;<br/><i><font color="Green">E91</font></i>:  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
 <a href="radix_5.html#K7" target="_self">FmaxDigit</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="txt" href="radix_5.html#E7:29"><i><font color="Green">E44</font></i></a>, <a class="txt" href="radix_5.html#E2:29"><i><font color="Green">E21</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="txt" href="radix_5.html#E6:29"><i><font color="Green">E43</font></i></a>, <a class="txt" href="radix_5.html#E1:29"><i><font color="Green">E20</font></i></a></i>
;<br/><i><font color="Green">E93</font></i>:  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> = 
 <a href="radix_5.html#K5" target="_self">FminDigit</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">m</font>
<b>by </b><i>, <a class="txt" href="radix_5.html#E7:29"><i><font color="Green">E44</font></i></a>, </i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="radix_5.html#T7" target="_self">Th7</a>, <a class="ref" href="radix_5.html#T1" target="_self">Th1</a></i>
;<br/> <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="radix_1.html#K14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="radix_1.html#K12">SD_Add_Data</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="radix_5.html#T5" target="_self">Th5</a>, <a class="ref" href="radix_5.html#T8" target="_self">Th8</a>, <a class="txt" href="radix_5.html#E7:29"><i><font color="Green">E44</font></i></a>, , , <a class="txt" href="radix_5.html#E8:29"><i><font color="Green">E47</font></i></a>, <a class="ref" href="radix_5.html#T1" target="_self">Th1</a>, <a class="ref" href="radix_1.html#T21">RADIX_1:21</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E7:29_4_2_1_2_1_1_1_1_1_1"/>
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="radix_1.html#K14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="radix_5.html#E9:29"><i><font color="Green">E48</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_2_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_4_2_1_2_1_1_1_1_1_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:29_4_2_1_2_1_1_1_1_1_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="radix_5.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan5b.html#T1">JORDAN5B:1</a></i>;<br/><a NAME="E3:29_4_2_1_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="radix_5.html#E7:29"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:29_4_2_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1 <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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:29_4_2_1_2_1_1_1_1_1_2"/>
<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="txt" href="radix_5.html#E13"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E6:29_4_2_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<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> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:29_4_2_1_2_1_1_1_1_1_2"/><i><font color="Green">E98</font></i>: 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="txt" href="radix_5.html#E5:29"><i><font color="Green">E36</font></i></a>, <a class="txt" href="radix_5.html#E7:29"><i><font color="Green">E44</font></i></a>, <a class="txt" href="radix_5.html#E1:29"><i><font color="Green">E20</font></i></a></i>;<br/><i><font color="Green">E99</font></i>:  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
 <a href="radix_5.html#K5" target="_self">FminDigit</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="radix_5.html#E8:29"><i><font color="Green">E47</font></i></a>, </i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="txt" href="radix_5.html#E7:29"><i><font color="Green">E44</font></i></a>, , <a class="ref" href="radix_5.html#T1" target="_self">Th1</a></i>
;<br/><i><font color="Green">E100</font></i>:  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
 <a href="radix_5.html#K7" target="_self">FmaxDigit</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="radix_5.html#E8:29"><i><font color="Green">E47</font></i></a>, <a class="txt" href="radix_5.html#E2:29"><i><font color="Green">E21</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="txt" href="radix_5.html#E7:29"><i><font color="Green">E44</font></i></a>, , <a class="txt" href="radix_5.html#E1:29"><i><font color="Green">E20</font></i></a></i>
;<br/><a NAME="E10:29_4_2_1_2_1_1_1_1_1_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="radix_5.html#E7:29"><i><font color="Green">E44</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/> <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="radix_1.html#K14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="radix_1.html#K12">SD_Add_Data</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 0</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_1.html#K11">SD_Add_Carry</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="radix_5.html#T5" target="_self">Th5</a>, <a class="ref" href="radix_5.html#T8" target="_self">Th8</a>, <a class="txt" href="radix_5.html#E7:29"><i><font color="Green">E44</font></i></a>, <a class="txt" href="radix_5.html#E9:29"><i><font color="Green">E48</font></i></a>, , , <a class="ref" href="radix_5.html#T1" target="_self">Th1</a></i>
<br/>.= 
0 <a href="nat_1.html#K1">+</a> 0
<b>by </b><i><a class="ref" href="radix_1.html#T21">RADIX_1:21</a>, <a class="ref" href="radix_1.html#T22">RADIX_1:22</a></i>
;<br/><b>hence </b><a NAME="E12:29_4_2_1_2_1_1_1_1_1_2"/>
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="radix_1.html#K14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="radix_5.html#T10" target="_self">Th10</a>, <a class="txt" href="radix_5.html#E1:29_4"><i><font color="Green">E51</font></i></a>, <a class="ref" href="radix_5.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:29_4_2_1_2_1_1_1"/>
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="radix_1.html#K14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:29_4_2_1_2_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:29_4_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E102</font></i>: 
<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="E3:29_4_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E103</font></i>: 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_4_2_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_2_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_4_2_1_2_1_1_2_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 </span><b>by </b><i><a class="txt" href="radix_5.html#E12"><i><font color="Green">Lemma39</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_2_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:29_4_2_1_2_1_1_2_1_1_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:29_4_2_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E105</font></i>: 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="radix_5.html#T8" target="_self">Th8</a>, <a class="ref" href="radix_5.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E3:29_4_2_1_2_1_1_2_1_1_1"/><i><font color="Green">E106</font></i>: 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="txt" href="radix_5.html#E5:29"><i><font color="Green">E36</font></i></a>, , <a class="txt" href="radix_5.html#E1:29"><i><font color="Green">E20</font></i></a></i>;<br/><a NAME="E4:29_4_2_1_2_1_1_2_1_1_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/> <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="radix_1.html#K14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="radix_1.html#K12">SD_Add_Data</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 0</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_1.html#K11">SD_Add_Carry</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="radix_5.html#T5" target="_self">Th5</a>, <a class="txt" href="radix_5.html#E13"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="radix_5.html#T12" target="_self">Th12</a>, <a class="txt" href="radix_5.html#E6:29"><i><font color="Green">E43</font></i></a>, <a class="txt" href="radix_5.html#E7:29"><i><font color="Green">E44</font></i></a>, <a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>
<br/>.= 
0 <a href="nat_1.html#K1">+</a> 0
<b>by </b><i><a class="ref" href="radix_1.html#T21">RADIX_1:21</a>, <a class="ref" href="radix_1.html#T22">RADIX_1:22</a></i>
;<br/><b>hence </b><a NAME="E6:29_4_2_1_2_1_1_2_1_1_1"/>
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="radix_1.html#K14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="radix_5.html#T10" target="_self">Th10</a>, , <a class="ref" href="radix_5.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_2_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:29_4_2_1_2_1_1_2_1_1_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:29_4_2_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E109</font></i>: 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="radix_5.html#T8" target="_self">Th8</a>, <a class="ref" href="radix_5.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E3:29_4_2_1_2_1_1_2_1_1_2"/><i><font color="Green">E110</font></i>: 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="txt" href="radix_5.html#E5:29"><i><font color="Green">E36</font></i></a>, , <a class="txt" href="radix_5.html#E1:29"><i><font color="Green">E20</font></i></a></i>;<br/><a NAME="E4:29_4_2_1_2_1_1_2_1_1_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/> <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="radix_1.html#K14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="radix_1.html#K12">SD_Add_Data</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
<b>by </b><i><a class="ref" href="radix_5.html#T5" target="_self">Th5</a>, <a class="txt" href="radix_5.html#E13"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="radix_5.html#T12" target="_self">Th12</a>, <a class="txt" href="radix_5.html#E6:29"><i><font color="Green">E43</font></i></a>, <a class="txt" href="radix_5.html#E7:29"><i><font color="Green">E44</font></i></a>, <a class="ref" href="radix_1.html#T21">RADIX_1:21</a>, <a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E6:29_4_2_1_2_1_1_2_1_1_2"/>
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="radix_1.html#K14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="radix_5.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="radix_5.html#T10" target="_self">Th10</a>, , <a class="ref" href="radix_5.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:29_4_2_1_2_1_1_2"/>
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="radix_1.html#K14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:29_4_2_1_2"/>
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="radix_1.html#K14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:29_4"/>
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="radix_1.html#K14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:29"/>
 <a href="radix_1.html#K8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <span class="p2">(<span class="default"><a href="radix_5.html#K6" target="_self">Fmin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" target="_self">Fmax</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="radix_5.html#E4:29"><i><font color="Green">E30</font></i></a>, </i>;<br/>


</div>
