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

<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 implies  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:29"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E2:29"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span></span><br/>

<a NAME="E3:29"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E4:29"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E5:29"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E27">A5</font></span>:  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) = 
 <a href="radix_5.html#K5" title="RADIX_5:func.5">FminDigit</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_5.html#D6" target="_self" title="RADIX_5:def.6">Def6</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E4:29"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="radix_5.html#D5" target="_self" title="RADIX_5:def.5">Def5</a></span>
;<br/>
<a NAME="E7:29"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="finseq_2.html#T8" title="FINSEQ_2:th.8">FINSEQ_2:8</a></span>;<br/>
<a NAME="E8:29"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E9:29"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/> <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Olive" title="b1">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Olive" title="b1">i</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>) )</span><br/>

<a NAME="E1:29_2"/><span class="lab"><font color="Green" title="E31">A9</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:29_2"/><span class="lab"><font color="Green" title="E32">A10</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)</span><br/>

<span class="kw">then </span><span class="lab"><font color="Green" title="E33">A11</font></span>:  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>) = 
 <a href="radix_1.html#K13" title="RADIX_1:func.13">Add</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c3">k</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_1.html#D14" title="RADIX_1:def.14">RADIX_1:def 14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K12" title="RADIX_1:func.12">SD_Add_Data</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p4">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p4">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K11" title="RADIX_1:func.11">SD_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p4">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p4">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E2:29_2"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="radix_1.html#D13" title="RADIX_1:def.13">RADIX_1:def 13</a></span>
;<br/>
<a NAME="E4:29_2"/><span class="lab"><font color="Green" title="E34">A12</font></span>: 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_5.html#K5" title="RADIX_5:func.5">FminDigit</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">i</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_2"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="radix_5.html#D6" target="_self" title="RADIX_5:def.6">Def6</a></span>;<br/>
<a NAME="E5:29_2"/><span class="lab"><font color="Green" title="E35">A13</font></span>: 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_5.html#K5" title="RADIX_5:func.5">FminDigit</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">i</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_2"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="radix_5.html#D6" target="_self" title="RADIX_5:def.6">Def6</a></span>;<br/>
<a NAME="E6:29_2"/><span class="lab"><font color="Green" title="E36">A14</font></span>: 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_5.html#K7" title="RADIX_5:func.7">FmaxDigit</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">i</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_2"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="radix_5.html#D8" target="_self" title="RADIX_5:def.8">Def8</a></span>;<br/>
<a NAME="E7:29_2"/><span class="lab"><font color="Green" title="E37">A15</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_2"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:29_2_2_1"/>
( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_2_2_1_1"/><span class="lab"><font color="Green" title="E38">A16</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_2_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:29_2_2_1_1_1_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_2_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A17</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)</span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A18</font></span>:  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) = 
 <a href="radix_5.html#K5" title="RADIX_5:func.5">FminDigit</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_2"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="radix_5.html#D6" target="_self" title="RADIX_5:def.6">Def6</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="radix_5.html#D5" target="_self" title="RADIX_5:def.5">Def5</a></span>
;<br/><span class="lab"><font color="Green" title="E41">A19</font></span>:  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) = 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c2">m</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_2_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A17</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
<br/>.= 
 <a href="radix_5.html#K7" title="RADIX_5:func.7">FmaxDigit</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">m</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="radix_5.html#D8" target="_self" title="RADIX_5:def.8">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="radix_5.html#D7" target="_self" title="RADIX_5:def.7">Def7</a></span>
;<br/><span class="lab"><font color="Green" title="E42">A20</font></span>:  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) = 
 <a href="radix_5.html#K7" title="RADIX_5:func.7">FmaxDigit</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_2"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="txt" href="#E1:29_2_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A17</font></span></a>, <a class="ref" href="radix_5.html#D8" target="_self" title="RADIX_5:def.8">Def8</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E1:29_2"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="ref" href="radix_5.html#D7" target="_self" title="RADIX_5:def.7">Def7</a></span>
;<br/><span class="lab"><font color="Green" title="E43">A21</font></span>:  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) = 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c2">m</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_2_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A17</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
<br/>.= 
 <a href="radix_5.html#K5" title="RADIX_5:func.5">FminDigit</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">m</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="radix_5.html#D6" target="_self" title="RADIX_5:def.6">Def6</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="radix_5.html#D5" target="_self" title="RADIX_5:def.5">Def5</a></span>
;<br/> <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) = 
 <a href="radix_5.html#K5" title="RADIX_5:func.5">FminDigit</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_2"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="txt" href="#E1:29_2_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A17</font></span></a>, <a class="ref" href="radix_5.html#D6" target="_self" title="RADIX_5:def.6">Def6</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E1:29_2"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="ref" href="radix_5.html#D5" target="_self" title="RADIX_5:def.5">Def5</a></span>
;<br/><span class="kw">then </span> <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>) = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K11" title="RADIX_1:func.11">SD_Add_Carry</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_2"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E4:29_2_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="txt" href="#E5:29_2_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="txt" href="#E3:29_2_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A19</font></span></a>, <a class="ref" href="radix_1.html#T19" title="RADIX_1:th.19">RADIX_1:19</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="radix_5.html#T10" target="_self" title="RADIX_5:th.10">Th10</a></span>
;<br/><span class="kw">hence </span><a NAME="E8:29_2_2_1_1_1_1_1"/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_2_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_2_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E39">A22</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)</span><br/><div class="add"><a NAME="E2:29_2_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_2_2_1_1"><span class="lab"><font color="Green" title="E38">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:29_2_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T14" title="XREAL_1:th.14">XREAL_1:14</a></span>;<br/><a NAME="E4:29_2_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A23</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><a NAME="E5:29_2_2_1_1_1_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_2_2_1_1"><span class="lab"><font color="Green" title="E38">A16</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:29_2_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:29"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:29_2_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A24</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_2"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="radix_5.html#T2" target="_self" title="RADIX_5:th.2">Th2</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A25</font></span>:  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) = 
 <a href="radix_5.html#K5" title="RADIX_5:func.5">FminDigit</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_5.html#D6" target="_self" title="RADIX_5:def.6">Def6</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E4:29_2_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E40">A23</font></span></a>, <a class="ref" href="radix_5.html#D5" target="_self" title="RADIX_5:def.5">Def5</a></span>
;<br/><a NAME="E9:29_2_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E43">A26</font></span>: 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E4:29_2"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E1:29_2_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E39">A22</font></span></a>, <a class="ref" href="radix_5.html#D5" target="_self" title="RADIX_5:def.5">Def5</a></span>;<br/><span class="lab"><font color="Green" title="E44">A27</font></span>:  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) = 
 <a href="radix_5.html#K7" title="RADIX_5:func.7">FmaxDigit</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_2_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E41">A24</font></span></a>, <a class="ref" href="radix_5.html#D8" target="_self" title="RADIX_5:def.8">Def8</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E4:29_2_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E40">A23</font></span></a>, <a class="ref" href="radix_5.html#D7" target="_self" title="RADIX_5:def.7">Def7</a></span>
;<br/><a NAME="E11:29_2_2_1_1_1_1_2"/>
(  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E1:29_2"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="txt" href="#E5:29_2"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E6:29_2"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="txt" href="#E1:29_2_2_1_1"><span class="lab"><font color="Green" title="E38">A16</font></span></a>, <a class="ref" href="radix_5.html#D5" target="_self" title="RADIX_5:def.5">Def5</a>, <a class="ref" href="radix_5.html#D7" target="_self" title="RADIX_5:def.7">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E12:29_2_2_1_1_1_1_2"/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_2"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E8:29_2_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E42">A25</font></span></a>, <a class="txt" href="#E10:29_2_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E44">A27</font></span></a>, <a class="txt" href="#E9:29_2_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E43">A26</font></span></a>, <a class="ref" href="radix_1.html#T18" title="RADIX_1:th.18">RADIX_1:18</a>, <a class="ref" href="radix_1.html#T19" title="RADIX_1:th.19">RADIX_1:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:29_2_2_1_1"/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_2_2_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)</span><br/><div class="add"><a NAME="E2:29_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A28</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_2_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:29_2_2_1_2_1_1"/>
( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_2"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_2_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E39">A29</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)</span><br/><div class="add"><a NAME="E2:29_2_2_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A30</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_2_2_1_2"><span class="lab"><font color="Green" title="E38">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:29_2_2_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A31</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5b.html#T1" title="JORDAN5B:th.1">JORDAN5B:1</a></span>;<br/><a NAME="E4:29_2_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E42">A32</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E2:29_2_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E40">A30</font></span></a>, <a class="ref" href="radix_5.html#T2" target="_self" title="RADIX_5:th.2">Th2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_2_1_2_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_2_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:29_2_2_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> or <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_2_2_1_2"><span class="lab"><font color="Green" title="E38">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_2_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_2_2_1_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E43">A33</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)</span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A34</font></span>:  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) = 
 <a href="radix_5.html#K7" title="RADIX_5:func.7">FmaxDigit</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_2_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E42">A32</font></span></a>, <a class="ref" href="radix_5.html#D8" target="_self" title="RADIX_5:def.8">Def8</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E3:29_2_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E41">A31</font></span></a>, <a class="ref" href="radix_5.html#D7" target="_self" title="RADIX_5:def.7">Def7</a></span>
;<br/><a NAME="E3:29_2_2_1_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A35</font></span>: 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E6:29_2"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="txt" href="#E1:29_2_2_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A33</font></span></a>, <a class="ref" href="radix_5.html#D7" target="_self" title="RADIX_5:def.7">Def7</a></span>;<br/><span class="lab"><font color="Green" title="E46">A36</font></span>:  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) = 
 <a href="radix_5.html#K5" title="RADIX_5:func.5">FminDigit</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">m</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_2"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="txt" href="#E1:29_2_2_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A33</font></span></a>, <a class="ref" href="radix_5.html#D6" target="_self" title="RADIX_5:def.6">Def6</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E1:29_2"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="ref" href="radix_5.html#D5" target="_self" title="RADIX_5:def.5">Def5</a></span>
;<br/> <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) = 
 <a href="radix_5.html#K5" title="RADIX_5:func.5">FminDigit</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_2_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E42">A32</font></span></a>, <a class="txt" href="#E1:29_2_2_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A33</font></span></a>, <a class="ref" href="radix_5.html#D6" target="_self" title="RADIX_5:def.6">Def6</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E3:29_2_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E41">A31</font></span></a>, <a class="ref" href="radix_5.html#D5" target="_self" title="RADIX_5:def.5">Def5</a></span>
;<br/><span class="kw">then </span> <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>) = 
<span class="p1">(<span class="default"><a href="radix_1.html#K12" title="RADIX_1:func.12">SD_Add_Data</a> (<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E3:29_2"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E5:29_2"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E1:29_2_2_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A33</font></span></a>, <a class="txt" href="#E3:29_2_2_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A35</font></span></a>, <a class="txt" href="#E2:29_2_2_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A34</font></span></a>, <a class="ref" href="radix_5.html#D5" target="_self" title="RADIX_5:def.5">Def5</a>, <a class="ref" href="radix_1.html#T18" title="RADIX_1:th.18">RADIX_1:18</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="radix_5.html#T11" target="_self" title="RADIX_5:th.11">Th11</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:29_2_2_1_2_1_1_1_1_1_1"/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_2_2_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A36</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_2_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_2_2_1_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E43">A37</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)</span><br/><div class="add"><a NAME="E2:29_2_2_1_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E44">A38</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_2_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E39">A29</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/><a NAME="E3:29_2_2_1_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A39</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_2"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="ref" href="jordan5b.html#T1" title="JORDAN5B:th.1">JORDAN5B:1</a></span>;<br/><a NAME="E4:29_2_2_1_2_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_2_2_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E43">A37</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E5:29_2_2_1_2_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:29_2_2_1_2_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A40</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_2_2_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E44">A38</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A41</font></span>:  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) = 
 <a href="radix_5.html#K7" title="RADIX_5:func.7">FmaxDigit</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_5.html#D8" target="_self" title="RADIX_5:def.8">Def8</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E1:29_2_2_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E43">A37</font></span></a>, <a class="txt" href="#E3:29_2_2_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E45">A39</font></span></a>, <a class="ref" href="radix_5.html#D7" target="_self" title="RADIX_5:def.7">Def7</a></span>
;<br/><a NAME="E8:29_2_2_1_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E48">A42</font></span>: 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E6:29_2"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="txt" href="#E1:29_2_2_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E43">A37</font></span></a>, <a class="ref" href="radix_5.html#D7" target="_self" title="RADIX_5:def.7">Def7</a></span>;<br/><a NAME="E9:29_2_2_1_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E49">A43</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_2_2_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E43">A37</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/> <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) = 
 <a href="radix_5.html#K5" title="RADIX_5:func.5">FminDigit</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29_2_2_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E46">A40</font></span></a>, <a class="ref" href="radix_5.html#D6" target="_self" title="RADIX_5:def.6">Def6</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E1:29_2_2_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E43">A37</font></span></a>, <a class="txt" href="#E3:29_2_2_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E45">A39</font></span></a>, <a class="ref" href="radix_5.html#D5" target="_self" title="RADIX_5:def.5">Def5</a></span>
;<br/><span class="kw">then </span> <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>) = 
<span class="p1">(<span class="default"><a href="radix_1.html#K12" title="RADIX_1:func.12">SD_Add_Data</a> (<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K11" title="RADIX_1:func.11">SD_Add_Carry</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E3:29_2"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E5:29_2"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E1:29_2_2_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E43">A37</font></span></a>, <a class="txt" href="#E8:29_2_2_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E48">A42</font></span></a>, <a class="txt" href="#E7:29_2_2_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E47">A41</font></span></a>, <a class="ref" href="radix_5.html#D5" target="_self" title="RADIX_5:def.5">Def5</a></span>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_1.html#T18" title="RADIX_1:th.18">RADIX_1:18</a>, <a class="ref" href="radix_1.html#T19" title="RADIX_1:th.19">RADIX_1:19</a></span>
;<br/><span class="kw">hence </span><a NAME="E12:29_2_2_1_2_1_1_1_1_1_2"/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E4:29_2"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E9:29_2_2_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E49">A43</font></span></a>, <a class="ref" href="radix_5.html#D5" target="_self" title="RADIX_5:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:29_2_2_1_2_1_1_1"/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_2_2_1_2_1_1_2"/><span class="lab"><font color="Green" title="E39">A44</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)</span><br/><div class="add"><a NAME="E2:29_2_2_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A45</font></span>: 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T8" title="NAT_2:th.8">NAT_2:8</a>, <a class="ref" href="radix_1.html#D3" title="RADIX_1:def.3">RADIX_1:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_2_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:29_2_2_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_2_2_1_2"><span class="lab"><font color="Green" title="E38">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_2_2_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E41">A46</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)</span><br/><div class="add"><a NAME="E2:29_2_2_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
(  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E5:29_2"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E6:29_2"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="radix_5.html#D5" target="_self" title="RADIX_5:def.5">Def5</a>, <a class="ref" href="radix_5.html#D7" target="_self" title="RADIX_5:def.7">Def7</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A47</font></span>:  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>) = 
<span class="p1">(<span class="default"><a href="radix_1.html#K12" title="RADIX_1:func.12">SD_Add_Data</a> (<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K11" title="RADIX_1:func.11">SD_Add_Carry</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_2"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E1:29_2_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E39">A44</font></span></a>, <a class="txt" href="#E2:29_2_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E40">A45</font></span></a>, <a class="ref" href="nat_2.html#T8" title="NAT_2:th.8">NAT_2:8</a>, <a class="ref" href="radix_1.html#D3" title="RADIX_1:def.3">RADIX_1:def 3</a></span>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_1.html#T18" title="RADIX_1:th.18">RADIX_1:18</a>, <a class="ref" href="radix_1.html#T19" title="RADIX_1:th.19">RADIX_1:19</a></span>
;<br/><a NAME="E4:29_2_2_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_2_2_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E41">A46</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E5:29_2_2_1_2_1_1_2_1_1_1"/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E4:29_2"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E3:29_2_2_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E42">A47</font></span></a>, <a class="ref" href="radix_5.html#D5" target="_self" title="RADIX_5:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_2_2_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E41">A48</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)</span><br/><div class="add"><a NAME="E2:29_2_2_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
(  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E5:29_2"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E6:29_2"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="radix_5.html#D5" target="_self" title="RADIX_5:def.5">Def5</a>, <a class="ref" href="radix_5.html#D7" target="_self" title="RADIX_5:def.7">Def7</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A49</font></span>:  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>) = 
<span class="p1">(<span class="default"><a href="radix_1.html#K12" title="RADIX_1:func.12">SD_Add_Data</a> (<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_2"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E1:29_2_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E39">A44</font></span></a>, <a class="txt" href="#E2:29_2_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E40">A45</font></span></a>, <a class="ref" href="nat_2.html#T8" title="NAT_2:th.8">NAT_2:8</a>, <a class="ref" href="radix_1.html#T18" title="RADIX_1:th.18">RADIX_1:18</a>, <a class="ref" href="radix_1.html#D3" title="RADIX_1:def.3">RADIX_1:def 3</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="radix_5.html#T11" target="_self" title="RADIX_5:th.11">Th11</a></span>
;<br/><a NAME="E4:29_2_2_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_2_2_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E41">A48</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E5:29_2_2_1_2_1_1_2_1_1_2"/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E4:29_2"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E3:29_2_2_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E42">A49</font></span></a>, <a class="ref" href="radix_5.html#D5" target="_self" title="RADIX_5:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:29_2_2_1_2_1_1_2"/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:29_2_2_1_2"/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:29_2"/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E31">A50</font></span>:  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p1">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) = 
 <a href="radix_5.html#K7" title="RADIX_5:func.7">FmaxDigit</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="radix_5.html#D8" target="_self" title="RADIX_5:def.8">Def8</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E4:29"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="radix_5.html#D7" target="_self" title="RADIX_5:def.7">Def7</a></span>
;<br/>
<span class="p1">(<span class="default"><a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p3">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K11" title="RADIX_1:func.11">SD_Add_Carry</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p5">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<span class="p5">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="radix_2.html#T11" title="RADIX_2:th.11">RADIX_2:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p3">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E10:29"><span class="lab"><font color="Green" title="E31">A50</font></span></a>, <a class="ref" href="radix_1.html#T18" title="RADIX_1:th.18">RADIX_1:18</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:29"/>
 <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p2">(<span class="default"><a href="radix_5.html#K6" title="RADIX_5:func.6">Fmin</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p2">(<span class="default"><a href="radix_5.html#K8" title="RADIX_5:func.8">Fmax</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="ref" href="radix_5.html#T12" target="_self" title="RADIX_5:th.12">Th12</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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