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<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="Red">::  thesis: </font></span></a><span class="hide"> ( 3 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> iff 3 <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> <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><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">p</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="E2:14"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">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="E3:14"/><span class="lab"><font color="Green" title="E10">A2</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="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c2">p</font> holds <br/><font color="Maroon" title="c2">p</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="E4:14"/><span class="lab"><font color="Green" title="E11">A3</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">p</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">reconsider </span><font color="Maroon" title="c3">dop</font> =  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c2">p</font> 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="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> $1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3;<br/>
<a NAME="E6:14"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">dop</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">k</font>,<font color="Olive" title="b2">x</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">p9</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="E8:14"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">dop</font> &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">dop</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c4">p9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</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="#E6:14"><span class="lab"><font color="Green" title="E12">A4</font></span></a>);<br/></span>
<div><span class="lab"><font color="Green" title="E14">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c2">p</font> implies <font color="Maroon" title="c4">p9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">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="Maroon" title="c5">i</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3 )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c6">dz</font> = <span class="p1">(<span class="default">10 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a>, <a class="ref" href="newton.html#T102" title="NEWTON:th.102">NEWTON:102</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">dz</font> = <font color="Maroon" title="c6">dz</font> 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="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E3:14_1"/>
( 3 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 3 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 3 &amp; 9 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c7">dz</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T9" target="_self" title="NUMERAL1:th.9">Th9</a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/><a NAME="E4:14_1"/><span class="kw">then </span>
3 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c7">dz</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></span>;<br/><a NAME="E5:14_1"/><span class="kw">then </span>
3 <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> <font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c7">dz</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></span>;<br/><a NAME="E6:14_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c5">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c7">dz</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3 <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/><span class="kw">assume </span><a NAME="E7:14_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">p9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">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="Maroon" title="c5">i</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3</span><br/><span class="kw">hence </span><font color="Maroon" title="c4">p9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:14"><span class="lab"><font color="Green" title="E13">A5</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c5">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">dz</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E7:14_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a></span>
<br/>.= 
<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="c5">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c7">dz</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><span class="p3">(<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="c5">i</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3

<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><span class="p3">(<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="c5">i</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:14_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="nat_d.html#T22" title="NAT_D:th.22">NAT_D:22</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="c5">i</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3

;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E10:14"/>
( 3 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> implies 3 <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> <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 3 <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> <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> implies 3 <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="14_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:14_2"/>
3 <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="Red">::  thesis: </font></span></a><span class="hide"> 3 <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> <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><br/>

<a NAME="E2:14_2"/><span class="kw">then </span>
3 <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="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="numeral1.html#T8" target="_self" title="NUMERAL1:th.8">Th8</a></span>;<br/>
<a NAME="E3:14_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3 <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/>
<a NAME="E4:14_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c4">p9</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3 <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="#E8:14"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="numeral1.html#T6" target="_self" title="NUMERAL1:th.6">Th6</a></span>;<br/>
<a NAME="E5:14_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3 <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="#E2:14"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E8:14"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E9:14"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="numeral1.html#T6" target="_self" title="NUMERAL1:th.6">Th6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:14_2"/>
3 <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> <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="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E11:14"/>
3 <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> <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 3 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font></span><br/>

<a NAME="E12:14"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3 <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/>
<a NAME="E13:14"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c4">p9</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3 <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="#E2:14"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E8:14"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E9:14"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="numeral1.html#T6" target="_self" title="NUMERAL1:th.6">Th6</a></span>;<br/>
<a NAME="E14:14"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3 <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="#E8:14"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="numeral1.html#T6" target="_self" title="NUMERAL1:th.6">Th6</a></span>;<br/>
<a NAME="E15:14"/><span class="kw">then </span>
3 <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="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:14"/>
3 <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="txt" href="#E4:14"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="numeral1.html#T8" target="_self" title="NUMERAL1:th.8">Th8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
