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

<span class="kw">let </span><font color="Maroon" title="c1">n</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"> ( 5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> iff 5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">d</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E2:12"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:12"/><span class="lab"><font color="Green" title="E9">A1</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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">d</font> holds <br/><font color="Maroon" title="c2">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default">10 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Olive" title="b1">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:12"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
 <a href="numeral1.html#K1" title="NUMERAL1:func.1">value</a> (<span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span>,10) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c2">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#D1" target="_self" title="NUMERAL1:def.1">Def1</a></span>;<br/>
<span class="kw">thus </span><a NAME="E5:12"/>
( 5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> implies 5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> implies 5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:12_1"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T4" target="_self" title="NUMERAL1:th.4">Th7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:12_1"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> 5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">d</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:12_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:12_1"/><span class="lab"><font color="Green" title="E14">A6</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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">d</font> holds <br/><font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default">10 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Olive" title="b1">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:12_1"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
 <a href="numeral1.html#K1" title="NUMERAL1:func.1">value</a> (<span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span>,10) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c3">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#D1" target="_self" title="NUMERAL1:def.1">Def1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_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="E1:12_1_1"/>
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_1"><span class="lab"><font color="Green" title="E11">A3</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="12_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:12_1_1_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <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"> 5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><br/><div class="add"><a NAME="E2:12_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A9</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/><a NAME="E3:12_1_1_1"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c3">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E6:12_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th8</a></span>;<br/><a NAME="E4:12_1_1_1"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E1:12_1_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a></span>;<br/><a NAME="E5:12_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T34" title="AFINSQ_1:th.34">AFINSQ_1:34</a></span>;<br/><span class="kw">then </span> <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c3">d</font> = 
<font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.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="afinsq_2.html#T53" title="AFINSQ_2:th.53">AFINSQ_2:53</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default">10 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E5:12_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E2:12_1_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a></span>
;<br/><a NAME="E7:12_1_1_1"/><span class="kw">then </span>
5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_1_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="newton.html#T4" title="NEWTON:th.4">NEWTON:4</a></span>;<br/><span class="kw">hence </span><a NAME="E8:12_1_1_1"/>
5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 ; <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="12_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:12_1_1_2"/><span class="lab"><font color="Green" title="E16">A11</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <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"> 5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">k</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>;<br/><div><span class="lab"><font color="Green" title="E17">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">l</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="c5">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> implies <font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">l</font> )</span><br/><span class="kw">assume </span><a NAME="E1:12_1_1_2_1"/>
<font color="Maroon" title="c5">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
 ; <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="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">l</font></span><br/><a NAME="E2:12_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></span>;<br/><a NAME="E3:12_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/><a NAME="E4:12_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">l</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="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><span class="kw">hence </span><a NAME="E5:12_1_1_2_1"/>
<font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T34" title="AFINSQ_1:th.34">AFINSQ_1:34</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><a NAME="E4:12_1_1_2"/><span class="lab"><font color="Green" title="E18">A13</font></span>: 
 for <font color="Olive" title="b1">u</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">k</font> holds <br/> ex <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">x</font>]
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c5">d9</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E6:12_1_1_2"/><span class="lab"><font color="Green" title="E19">A14</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">d9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="stirl2_1.html#S5" title="STIRL2_1:sch.5">STIRL2_1:sch 5</a>(<a class="txt" href="#E4:12_1_1_2"><span class="lab"><font color="Green" title="E18">A13</font></span></a>);<br/></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">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="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">d9</font> implies 5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:12_1_1_2_2"/><span class="lab"><font color="Green" title="E20">A15</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">d9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> 5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/><a NAME="E2:12_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_1_1_2"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/><a NAME="E3:12_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E4:12_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A16</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/><font color="Maroon" title="c5">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">i</font> = 
<font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_1_1_2"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="txt" href="#E1:12_1_1_2_2"><span class="lab"><font color="Green" title="E20">A15</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default">10 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E4:12_1_1_2_2"><span class="lab"><font color="Green" title="E21">A16</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">10 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 10</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></span>
<br/>.= 
5 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p3">(<span class="default">10 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 2</span>)</span>

;<br/><span class="kw">hence </span><a NAME="E6:12_1_1_2_2"/>
5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</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><a NAME="E8:12_1_1_2"/><span class="kw">then </span>
5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c5">d9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T2" target="_self" title="NUMERAL1:th.2">Th5</a></span>;<br/><a NAME="E9:12_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A17</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c5">d9</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 5 <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="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/><div><span class="lab"><font color="Green" title="E21">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">l</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="c6">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">d9</font> implies <font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">l</font> )</span><br/><a NAME="E1:12_1_1_2_3"/><span class="lab"><font color="Green" title="E22">A19</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c6">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T34" title="AFINSQ_1:th.34">AFINSQ_1:34</a></span>;<br/><span class="kw">assume </span><a NAME="E2:12_1_1_2_3"/>
<font color="Maroon" title="c6">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">d9</font>
 ; <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="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">l</font></span><br/><span class="kw">hence </span><a NAME="E3:12_1_1_2_3"/>
<font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_1_1_2"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="txt" href="#E1:12_1_1_2_3"><span class="lab"><font color="Green" title="E22">A19</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> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">d</font> = 
<font color="Maroon" title="c4">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">d9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_1_1_2"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="ref" href="afinsq_1.html#T33" title="AFINSQ_1:th.33">AFINSQ_1:33</a></span>
;<br/><a NAME="E12:12_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c5">d9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_1_1_2"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="txt" href="#E10:12_1_1_2"><span class="lab"><font color="Green" title="E21">A18</font></span></a>, <a class="ref" href="afinsq_1.html#D3" title="AFINSQ_1:def.3">AFINSQ_1:def 3</a></span>;<br/><a NAME="E13:12_1_1_2"/><span class="kw">then </span>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c5">d9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T55" title="AFINSQ_2:th.55">AFINSQ_2:55</a></span>;<br/><a NAME="E14:12_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c5">d9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th8</a></span>;<br/><a NAME="E15:12_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c5">d9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T53" title="AFINSQ_2:th.53">AFINSQ_2:53</a></span>;<br/><a NAME="E16:12_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c5">d9</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 5 <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="#E2:12_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/><a NAME="E17:12_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 5 <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="#E9:12_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="ref" href="nat_d.html#T17" title="NAT_D:th.17">NAT_D:17</a></span>;<br/><a NAME="E18:12_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A20</font></span>: 
5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.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="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/><a NAME="E19:12_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_1_1_2"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/><a NAME="E20:12_1_1_2"/><span class="kw">then </span>
5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default">10 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E5:12_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E18:12_1_1_2"><span class="lab"><font color="Green" title="E22">A20</font></span></a></span>;<br/><a NAME="E21:12_1_1_2"/><span class="kw">then </span>
5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T4" title="NEWTON:th.4">NEWTON:4</a></span>;<br/><span class="kw">hence </span><a NAME="E22:12_1_1_2"/>
5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 ; <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">assume </span><a NAME="E6:12"/><span class="lab"><font color="Green" title="E11">A21</font></span>: 
5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> 5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">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="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">d</font> implies 5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:12_2"/><span class="lab"><font color="Green" title="E12">A22</font></span>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> 5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_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="E1:12_2_1"/>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:12_2_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> 5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">then </span><font color="Maroon" title="c2">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default">10 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E1:12_2"><span class="lab"><font color="Green" title="E12">A22</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T4" title="NEWTON:th.4">NEWTON:4</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:12_2_1_1"/>
5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12"><span class="lab"><font color="Green" title="E11">A21</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="12_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:12_2_1_2"/>
<font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> 5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:12_2_1_2"/><span class="lab"><font color="Green" title="E13">A23</font></span>: 
<font color="Maroon" title="c4">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><font color="Maroon" title="c2">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default">10 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E1:12_2"><span class="lab"><font color="Green" title="E12">A22</font></span></a>, <a class="txt" href="#E3:12_2_1_2"><span class="lab"><font color="Green" title="E13">A23</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">10 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 10</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></span>
<br/>.= 
5 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p3">(<span class="default">10 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 2</span>)</span>

;<br/><span class="kw">hence </span><a NAME="E5:12_2_1_2"/>
5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</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>
<a NAME="E8:12"/><span class="kw">then </span>
5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="numeral1.html#K1" title="NUMERAL1:func.1">value</a> (<span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Maroon" title="c1">n</font>,10)</span>)</span>,10)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="numeral1.html#T2" target="_self" title="NUMERAL1:th.2">Th5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:12"/>
5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th8</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>
