
theorem
Sierp36 3980,20
proof
  thus Sum digits(3980,10) = 20 by Th727;
  3980=199*20;
  hence 20 divides 3980 by INT_1:def 3;
  let m be Nat;
  assume A1: Sum digits(m,10) = 20 & 20 divides m;
  then consider j being Nat such that
  A2: m=20*j by NAT_D:def 3;
  assume m < 3980;
  then 20*j < 20*199 by A2;
  then j < 198+1 by XREAL_1:64;
  then j <= 198 by NAT_1:9;
  then j <= 99 or (j >= 99+1 & j <= 198) by NAT_1:9;
  then (j=0 or ... or j=99) or (j=100 or ... or j=198);
  then per cases;
  suppose j=0;
    then Sum digits(m,10) = 0 by A2,Th6;
    hence contradiction by A1;
  end;
  suppose j=1;
    then Sum digits(m,10) = 2 by A2,Th19;
    hence contradiction by A1;
  end;
  suppose j=2;
    then Sum digits(m,10) = 4 by A2,Th23;
    hence contradiction by A1;
  end;
  suppose j=3;
    then Sum digits(m,10) = 6 by A2,Th27;
    hence contradiction by A1;
  end;
  suppose j=4;
    then Sum digits(m,10) = 8 by A2,Th31;
    hence contradiction by A1;
  end;
  suppose j=5;
    then Sum digits(m,10) = 1 by A2,Th35;
    hence contradiction by A1;
  end;
  suppose j=6;
    then Sum digits(m,10) = 3 by A2,Th39;
    hence contradiction by A1;
  end;
  suppose j=7;
    then Sum digits(m,10) = 5 by A2,Th43;
    hence contradiction by A1;
  end;
  suppose j=8;
    then Sum digits(m,10) = 7 by A2,Th47;
    hence contradiction by A1;
  end;
  suppose j=9;
    then Sum digits(m,10) = 9 by A2,Th51;
    hence contradiction by A1;
  end;
  suppose j=10;
    then Sum digits(m,10) = 2 by A2,Th361;
    hence contradiction by A1;
  end;
  suppose j=11;
    then Sum digits(m,10) = 4 by A2,Th363;
    hence contradiction by A1;
  end;
  suppose j=12;
    then Sum digits(m,10) = 6 by A2,Th193;
    hence contradiction by A1;
  end;
  suppose j=13;
    then Sum digits(m,10) = 8 by A2,Th365;
    hence contradiction by A1;
  end;
  suppose j=14;
    then Sum digits(m,10) = 10 by A2,Th367;
    hence contradiction by A1;
  end;
  suppose j=15;
    then Sum digits(m,10) = 3 by A2,Th369;
    hence contradiction by A1;
  end;
  suppose j=16;
    then Sum digits(m,10) = 5 by A2,Th203;
    hence contradiction by A1;
  end;
  suppose j=17;
    then Sum digits(m,10) = 7 by A2,Th250;
    hence contradiction by A1;
  end;
  suppose j=18;
    then Sum digits(m,10) = 9 by A2,Th371;
    hence contradiction by A1;
  end;
  suppose j=19;
    then Sum digits(m,10) = 11 by A2,Th306;
    hence contradiction by A1;
  end;
  suppose j=20;
    then Sum digits(m,10) = 4 by A2,Th213;
    hence contradiction by A1;
  end;
  suppose j=21;
    then Sum digits(m,10) = 6 by A2,Th373;
    hence contradiction by A1;
  end;
  suppose j=22;
    then Sum digits(m,10) = 8 by A2,Th375;
    hence contradiction by A1;
  end;
  suppose j=23;
    then Sum digits(m,10) = 10 by A2,Th377;
    hence contradiction by A1;
  end;
  suppose j=24;
    then Sum digits(m,10) = 12 by A2,Th379;
    hence contradiction by A1;
  end;
  suppose j=25;
    then Sum digits(m,10) = 5 by A2,Th381;
    hence contradiction by A1;
  end;
  suppose j=26;
    then Sum digits(m,10) = 7 by A2,Th383;
    hence contradiction by A1;
  end;
  suppose j=27;
    then Sum digits(m,10) = 9 by A2,Th385;
    hence contradiction by A1;
  end;
  suppose j=28;
    then Sum digits(m,10) = 11 by A2,Th387;
    hence contradiction by A1;
  end;
  suppose j=29;
    then Sum digits(m,10) = 13 by A2,Th389;
    hence contradiction by A1;
  end;
  suppose j=30;
    then Sum digits(m,10) = 6 by A2,Th391;
    hence contradiction by A1;
  end;
  suppose j=31;
    then Sum digits(m,10) = 8 by A2,Th393;
    hence contradiction by A1;
  end;
  suppose j=32;
    then Sum digits(m,10) = 10 by A2,Th395;
    hence contradiction by A1;
  end;
  suppose j=33;
    then Sum digits(m,10) = 12 by A2,Th397;
    hence contradiction by A1;
  end;
  suppose j=34;
    then Sum digits(m,10) = 14 by A2,Th399;
    hence contradiction by A1;
  end;
  suppose j=35;
    then Sum digits(m,10) = 7 by A2,Th401;
    hence contradiction by A1;
  end;
  suppose j=36;
    then Sum digits(m,10) = 9 by A2,Th403;
    hence contradiction by A1;
  end;
  suppose j=37;
    then Sum digits(m,10) = 11 by A2,Th405;
    hence contradiction by A1;
  end;
  suppose j=38;
    then Sum digits(m,10) = 13 by A2,Th346;
    hence contradiction by A1;
  end;
  suppose j=39;
    then Sum digits(m,10) = 15 by A2,Th407;
    hence contradiction by A1;
  end;
  suppose j=40;
    then Sum digits(m,10) = 8 by A2,Th409;
    hence contradiction by A1;
  end;
  suppose j=41;
    then Sum digits(m,10) = 10 by A2,Th411;
    hence contradiction by A1;
  end;
  suppose j=42;
    then Sum digits(m,10) = 12 by A2,Th413;
    hence contradiction by A1;
  end;
  suppose j=43;
    then Sum digits(m,10) = 14 by A2,Th415;
    hence contradiction by A1;
  end;
  suppose j=44;
    then Sum digits(m,10) = 16 by A2,Th417;
    hence contradiction by A1;
  end;
  suppose j=45;
    then Sum digits(m,10) = 9 by A2,Th419;
    hence contradiction by A1;
  end;
  suppose j=46;
    then Sum digits(m,10) = 11 by A2,Th421;
    hence contradiction by A1;
  end;
  suppose j=47;
    then Sum digits(m,10) = 13 by A2,Th423;
    hence contradiction by A1;
  end;
  suppose j=48;
    then Sum digits(m,10) = 15 by A2,Th425;
    hence contradiction by A1;
  end;
  suppose j=49;
    then Sum digits(m,10) = 17 by A2,Th427;
    hence contradiction by A1;
  end;
  suppose j=50;
    then Sum digits(m,10) = 1 by A2,Th429;
    hence contradiction by A1;
  end;
  suppose j=51;
    then Sum digits(m,10) = 3 by A2,Th431;
    hence contradiction by A1;
  end;
  suppose j=52;
    then Sum digits(m,10) = 5 by A2,Th433;
    hence contradiction by A1;
  end;
  suppose j=53;
    then Sum digits(m,10) = 7 by A2,Th435;
    hence contradiction by A1;
  end;
  suppose j=54;
    then Sum digits(m,10) = 9 by A2,Th437;
    hence contradiction by A1;
  end;
  suppose j=55;
    then Sum digits(m,10) = 2 by A2,Th439;
    hence contradiction by A1;
  end;
  suppose j=56;
    then Sum digits(m,10) = 4 by A2,Th441;
    hence contradiction by A1;
  end;
  suppose j=57;
    then Sum digits(m,10) = 6 by A2,Th443;
    hence contradiction by A1;
  end;
  suppose j=58;
    then Sum digits(m,10) = 8 by A2,Th445;
    hence contradiction by A1;
  end;
  suppose j=59;
    then Sum digits(m,10) = 10 by A2,Th447;
    hence contradiction by A1;
  end;
  suppose j=60;
    then Sum digits(m,10) = 3 by A2,Th449;
    hence contradiction by A1;
  end;
  suppose j=61;
    then Sum digits(m,10) = 5 by A2,Th451;
    hence contradiction by A1;
  end;
  suppose j=62;
    then Sum digits(m,10) = 7 by A2,Th453;
    hence contradiction by A1;
  end;
  suppose j=63;
    then Sum digits(m,10) = 9 by A2,Th455;
    hence contradiction by A1;
  end;
  suppose j=64;
    then Sum digits(m,10) = 11 by A2,Th457;
    hence contradiction by A1;
  end;
  suppose j=65;
    then Sum digits(m,10) = 4 by A2,Th459;
    hence contradiction by A1;
  end;
  suppose j=66;
    then Sum digits(m,10) = 6 by A2,Th461;
    hence contradiction by A1;
  end;
  suppose j=67;
    then Sum digits(m,10) = 8 by A2,Th463;
    hence contradiction by A1;
  end;
  suppose j=68;
    then Sum digits(m,10) = 10 by A2,Th465;
    hence contradiction by A1;
  end;
  suppose j=69;
    then Sum digits(m,10) = 12 by A2,Th467;
    hence contradiction by A1;
  end;
  suppose j=70;
    then Sum digits(m,10) = 5 by A2,Th469;
    hence contradiction by A1;
  end;
  suppose j=71;
    then Sum digits(m,10) = 7 by A2,Th471;
    hence contradiction by A1;
  end;
  suppose j=72;
    then Sum digits(m,10) = 9 by A2,Th473;
    hence contradiction by A1;
  end;
  suppose j=73;
    then Sum digits(m,10) = 11 by A2,Th475;
    hence contradiction by A1;
  end;
  suppose j=74;
    then Sum digits(m,10) = 13 by A2,Th477;
    hence contradiction by A1;
  end;
  suppose j=75;
    then Sum digits(m,10) = 6 by A2,Th479;
    hence contradiction by A1;
  end;
  suppose j=76;
    then Sum digits(m,10) = 8 by A2,Th481;
    hence contradiction by A1;
  end;
  suppose j=77;
    then Sum digits(m,10) = 10 by A2,Th483;
    hence contradiction by A1;
  end;
  suppose j=78;
    then Sum digits(m,10) = 12 by A2,Th485;
    hence contradiction by A1;
  end;
  suppose j=79;
    then Sum digits(m,10) = 14 by A2,Th487;
    hence contradiction by A1;
  end;
  suppose j=80;
    then Sum digits(m,10) = 7 by A2,Th489;
    hence contradiction by A1;
  end;
  suppose j=81;
    then Sum digits(m,10) = 9 by A2,Th491;
    hence contradiction by A1;
  end;
  suppose j=82;
    then Sum digits(m,10) = 11 by A2,Th493;
    hence contradiction by A1;
  end;
  suppose j=83;
    then Sum digits(m,10) = 13 by A2,Th495;
    hence contradiction by A1;
  end;
  suppose j=84;
    then Sum digits(m,10) = 15 by A2,Th497;
    hence contradiction by A1;
  end;
  suppose j=85;
    then Sum digits(m,10) = 8 by A2,Th499;
    hence contradiction by A1;
  end;
  suppose j=86;
    then Sum digits(m,10) = 10 by A2,Th501;
    hence contradiction by A1;
  end;
  suppose j=87;
    then Sum digits(m,10) = 12 by A2,Th503;
    hence contradiction by A1;
  end;
  suppose j=88;
    then Sum digits(m,10) = 14 by A2,Th505;
    hence contradiction by A1;
  end;
  suppose j=89;
    then Sum digits(m,10) = 16 by A2,Th507;
    hence contradiction by A1;
  end;
  suppose j=90;
    then Sum digits(m,10) = 9 by A2,Th509;
    hence contradiction by A1;
  end;
  suppose j=91;
    then Sum digits(m,10) = 11 by A2,Th511;
    hence contradiction by A1;
  end;
  suppose j=92;
    then Sum digits(m,10) = 13 by A2,Th513;
    hence contradiction by A1;
  end;
  suppose j=93;
    then Sum digits(m,10) = 15 by A2,Th515;
    hence contradiction by A1;
  end;
  suppose j=94;
    then Sum digits(m,10) = 17 by A2,Th517;
    hence contradiction by A1;
  end;
  suppose j=95;
    then Sum digits(m,10) = 10 by A2,Th519;
    hence contradiction by A1;
  end;
  suppose j=96;
    then Sum digits(m,10) = 12 by A2,Th521;
    hence contradiction by A1;
  end;
  suppose j=97;
    then Sum digits(m,10) = 14 by A2,Th523;
    hence contradiction by A1;
  end;
  suppose j=98;
    then Sum digits(m,10) = 16 by A2,Th525;
    hence contradiction by A1;
  end;
  suppose j=99;
    then Sum digits(m,10) = 18 by A2,Th527;
    hence contradiction by A1;
  end;
  suppose j=100;
    then Sum digits(m,10) = 2 by A2,Th529;
    hence contradiction by A1;
  end;
  suppose j=101;
    then Sum digits(m,10) = 4 by A2,Th531;
    hence contradiction by A1;
  end;
  suppose j=102;
    then Sum digits(m,10) = 6 by A2,Th533;
    hence contradiction by A1;
  end;
  suppose j=103;
    then Sum digits(m,10) = 8 by A2,Th535;
    hence contradiction by A1;
  end;
  suppose j=104;
    then Sum digits(m,10) = 10 by A2,Th537;
    hence contradiction by A1;
  end;
  suppose j=105;
    then Sum digits(m,10) = 3 by A2,Th539;
    hence contradiction by A1;
  end;
  suppose j=106;
    then Sum digits(m,10) = 5 by A2,Th541;
    hence contradiction by A1;
  end;
  suppose j=107;
    then Sum digits(m,10) = 7 by A2,Th543;
    hence contradiction by A1;
  end;
  suppose j=108;
    then Sum digits(m,10) = 9 by A2,Th545;
    hence contradiction by A1;
  end;
  suppose j=109;
    then Sum digits(m,10) = 11 by A2,Th547;
    hence contradiction by A1;
  end;
  suppose j=110;
    then Sum digits(m,10) = 4 by A2,Th549;
    hence contradiction by A1;
  end;
  suppose j=111;
    then Sum digits(m,10) = 6 by A2,Th551;
    hence contradiction by A1;
  end;
  suppose j=112;
    then Sum digits(m,10) = 8 by A2,Th553;
    hence contradiction by A1;
  end;
  suppose j=113;
    then Sum digits(m,10) = 10 by A2,Th555;
    hence contradiction by A1;
  end;
  suppose j=114;
    then Sum digits(m,10) = 12 by A2,Th557;
    hence contradiction by A1;
  end;
  suppose j=115;
    then Sum digits(m,10) = 5 by A2,Th559;
    hence contradiction by A1;
  end;
  suppose j=116;
    then Sum digits(m,10) = 7 by A2,Th561;
    hence contradiction by A1;
  end;
  suppose j=117;
    then Sum digits(m,10) = 9 by A2,Th563;
    hence contradiction by A1;
  end;
  suppose j=118;
    then Sum digits(m,10) = 11 by A2,Th565;
    hence contradiction by A1;
  end;
  suppose j=119;
    then Sum digits(m,10) = 13 by A2,Th567;
    hence contradiction by A1;
  end;
  suppose j=120;
    then Sum digits(m,10) = 6 by A2,Th569;
    hence contradiction by A1;
  end;
  suppose j=121;
    then Sum digits(m,10) = 8 by A2,Th571;
    hence contradiction by A1;
  end;
  suppose j=122;
    then Sum digits(m,10) = 10 by A2,Th573;
    hence contradiction by A1;
  end;
  suppose j=123;
    then Sum digits(m,10) = 12 by A2,Th575;
    hence contradiction by A1;
  end;
  suppose j=124;
    then Sum digits(m,10) = 14 by A2,Th577;
    hence contradiction by A1;
  end;
  suppose j=125;
    then Sum digits(m,10) = 7 by A2,Th579;
    hence contradiction by A1;
  end;
  suppose j=126;
    then Sum digits(m,10) = 9 by A2,Th581;
    hence contradiction by A1;
  end;
  suppose j=127;
    then Sum digits(m,10) = 11 by A2,Th583;
    hence contradiction by A1;
  end;
  suppose j=128;
    then Sum digits(m,10) = 13 by A2,Th585;
    hence contradiction by A1;
  end;
  suppose j=129;
    then Sum digits(m,10) = 15 by A2,Th587;
    hence contradiction by A1;
  end;
  suppose j=130;
    then Sum digits(m,10) = 8 by A2,Th589;
    hence contradiction by A1;
  end;
  suppose j=131;
    then Sum digits(m,10) = 10 by A2,Th591;
    hence contradiction by A1;
  end;
  suppose j=132;
    then Sum digits(m,10) = 12 by A2,Th593;
    hence contradiction by A1;
  end;
  suppose j=133;
    then Sum digits(m,10) = 14 by A2,Th595;
    hence contradiction by A1;
  end;
  suppose j=134;
    then Sum digits(m,10) = 16 by A2,Th597;
    hence contradiction by A1;
  end;
  suppose j=135;
    then Sum digits(m,10) = 9 by A2,Th599;
    hence contradiction by A1;
  end;
  suppose j=136;
    then Sum digits(m,10) = 11 by A2,Th601;
    hence contradiction by A1;
  end;
  suppose j=137;
    then Sum digits(m,10) = 13 by A2,Th603;
    hence contradiction by A1;
  end;
  suppose j=138;
    then Sum digits(m,10) = 15 by A2,Th605;
    hence contradiction by A1;
  end;
  suppose j=139;
    then Sum digits(m,10) = 17 by A2,Th607;
    hence contradiction by A1;
  end;
  suppose j=140;
    then Sum digits(m,10) = 10 by A2,Th609;
    hence contradiction by A1;
  end;
  suppose j=141;
    then Sum digits(m,10) = 12 by A2,Th611;
    hence contradiction by A1;
  end;
  suppose j=142;
    then Sum digits(m,10) = 14 by A2,Th613;
    hence contradiction by A1;
  end;
  suppose j=143;
    then Sum digits(m,10) = 16 by A2,Th615;
    hence contradiction by A1;
  end;
  suppose j=144;
    then Sum digits(m,10) = 18 by A2,Th617;
    hence contradiction by A1;
  end;
  suppose j=145;
    then Sum digits(m,10) = 11 by A2,Th619;
    hence contradiction by A1;
  end;
  suppose j=146;
    then Sum digits(m,10) = 13 by A2,Th621;
    hence contradiction by A1;
  end;
  suppose j=147;
    then Sum digits(m,10) = 15 by A2,Th623;
    hence contradiction by A1;
  end;
  suppose j=148;
    then Sum digits(m,10) = 17 by A2,Th625;
    hence contradiction by A1;
  end;
  suppose j=149;
    then Sum digits(m,10) = 19 by A2,Th627;
    hence contradiction by A1;
  end;
  suppose j=150;
    then Sum digits(m,10) = 3 by A2,Th629;
    hence contradiction by A1;
  end;
  suppose j=151;
    then Sum digits(m,10) = 5 by A2,Th631;
    hence contradiction by A1;
  end;
  suppose j=152;
    then Sum digits(m,10) = 7 by A2,Th633;
    hence contradiction by A1;
  end;
  suppose j=153;
    then Sum digits(m,10) = 9 by A2,Th635;
    hence contradiction by A1;
  end;
  suppose j=154;
    then Sum digits(m,10) = 11 by A2,Th637;
    hence contradiction by A1;
  end;
  suppose j=155;
    then Sum digits(m,10) = 4 by A2,Th639;
    hence contradiction by A1;
  end;
  suppose j=156;
    then Sum digits(m,10) = 6 by A2,Th641;
    hence contradiction by A1;
  end;
  suppose j=157;
    then Sum digits(m,10) = 8 by A2,Th643;
    hence contradiction by A1;
  end;
  suppose j=158;
    then Sum digits(m,10) = 10 by A2,Th645;
    hence contradiction by A1;
  end;
  suppose j=159;
    then Sum digits(m,10) = 12 by A2,Th647;
    hence contradiction by A1;
  end;
  suppose j=160;
    then Sum digits(m,10) = 5 by A2,Th649;
    hence contradiction by A1;
  end;
  suppose j=161;
    then Sum digits(m,10) = 7 by A2,Th651;
    hence contradiction by A1;
  end;
  suppose j=162;
    then Sum digits(m,10) = 9 by A2,Th653;
    hence contradiction by A1;
  end;
  suppose j=163;
    then Sum digits(m,10) = 11 by A2,Th655;
    hence contradiction by A1;
  end;
  suppose j=164;
    then Sum digits(m,10) = 13 by A2,Th657;
    hence contradiction by A1;
  end;
  suppose j=165;
    then Sum digits(m,10) = 6 by A2,Th659;
    hence contradiction by A1;
  end;
  suppose j=166;
    then Sum digits(m,10) = 8 by A2,Th661;
    hence contradiction by A1;
  end;
  suppose j=167;
    then Sum digits(m,10) = 10 by A2,Th663;
    hence contradiction by A1;
  end;
  suppose j=168;
    then Sum digits(m,10) = 12 by A2,Th665;
    hence contradiction by A1;
  end;
  suppose j=169;
    then Sum digits(m,10) = 14 by A2,Th667;
    hence contradiction by A1;
  end;
  suppose j=170;
    then Sum digits(m,10) = 7 by A2,Th669;
    hence contradiction by A1;
  end;
  suppose j=171;
    then Sum digits(m,10) = 9 by A2,Th671;
    hence contradiction by A1;
  end;
  suppose j=172;
    then Sum digits(m,10) = 11 by A2,Th673;
    hence contradiction by A1;
  end;
  suppose j=173;
    then Sum digits(m,10) = 13 by A2,Th675;
    hence contradiction by A1;
  end;
  suppose j=174;
    then Sum digits(m,10) = 15 by A2,Th677;
    hence contradiction by A1;
  end;
  suppose j=175;
    then Sum digits(m,10) = 8 by A2,Th679;
    hence contradiction by A1;
  end;
  suppose j=176;
    then Sum digits(m,10) = 10 by A2,Th681;
    hence contradiction by A1;
  end;
  suppose j=177;
    then Sum digits(m,10) = 12 by A2,Th683;
    hence contradiction by A1;
  end;
  suppose j=178;
    then Sum digits(m,10) = 14 by A2,Th685;
    hence contradiction by A1;
  end;
  suppose j=179;
    then Sum digits(m,10) = 16 by A2,Th687;
    hence contradiction by A1;
  end;
  suppose j=180;
    then Sum digits(m,10) = 9 by A2,Th689;
    hence contradiction by A1;
  end;
  suppose j=181;
    then Sum digits(m,10) = 11 by A2,Th691;
    hence contradiction by A1;
  end;
  suppose j=182;
    then Sum digits(m,10) = 13 by A2,Th693;
    hence contradiction by A1;
  end;
  suppose j=183;
    then Sum digits(m,10) = 15 by A2,Th695;
    hence contradiction by A1;
  end;
  suppose j=184;
    then Sum digits(m,10) = 17 by A2,Th697;
    hence contradiction by A1;
  end;
  suppose j=185;
    then Sum digits(m,10) = 10 by A2,Th699;
    hence contradiction by A1;
  end;
  suppose j=186;
    then Sum digits(m,10) = 12 by A2,Th701;
    hence contradiction by A1;
  end;
  suppose j=187;
    then Sum digits(m,10) = 14 by A2,Th703;
    hence contradiction by A1;
  end;
  suppose j=188;
    then Sum digits(m,10) = 16 by A2,Th705;
    hence contradiction by A1;
  end;
  suppose j=189;
    then Sum digits(m,10) = 18 by A2,Th707;
    hence contradiction by A1;
  end;
  suppose j=190;
    then Sum digits(m,10) = 11 by A2,Th709;
    hence contradiction by A1;
  end;
  suppose j=191;
    then Sum digits(m,10) = 13 by A2,Th711;
    hence contradiction by A1;
  end;
  suppose j=192;
    then Sum digits(m,10) = 15 by A2,Th713;
    hence contradiction by A1;
  end;
  suppose j=193;
    then Sum digits(m,10) = 17 by A2,Th715;
    hence contradiction by A1;
  end;
  suppose j=194;
    then Sum digits(m,10) = 19 by A2,Th717;
    hence contradiction by A1;
  end;
  suppose j=195;
    then Sum digits(m,10) = 12 by A2,Th719;
    hence contradiction by A1;
  end;
  suppose j=196;
    then Sum digits(m,10) = 14 by A2,Th721;
    hence contradiction by A1;
  end;
  suppose j=197;
    then Sum digits(m,10) = 16 by A2,Th723;
    hence contradiction by A1;
  end;
  suppose j=198;
    then Sum digits(m,10) = 18 by A2,Th725;
    hence contradiction by A1;
  end;
end;
