
theorem
Sierp36 874,19
proof
  thus Sum digits(874,10) = 19 by Th358;
  874=46*19;
  hence 19 divides 874 by INT_1:def 3;
  let m be Nat;
  assume A1: Sum digits(m,10) = 19 & 19 divides m;
  then consider j being Nat such that
  A2: m=19*j by NAT_D:def 3;
  assume m < 874;
  then 19*j < 19*46 by A2;
  then j < 45+1 by XREAL_1:64;
  then j <= 45 by NAT_1:9;
  then j=0 or ... or j=45;
  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) = 10 by A2,Th280;
    hence contradiction by A1;
  end;
  suppose j=2;
    then Sum digits(m,10) = 11 by A2,Th282;
    hence contradiction by A1;
  end;
  suppose j=3;
    then Sum digits(m,10) = 12 by A2,Th284;
    hence contradiction by A1;
  end;
  suppose j=4;
    then Sum digits(m,10) = 13 by A2,Th286;
    hence contradiction by A1;
  end;
  suppose j=5;
    then Sum digits(m,10) = 14 by A2,Th288;
    hence contradiction by A1;
  end;
  suppose j=6;
    then Sum digits(m,10) = 6 by A2,Th290;
    hence contradiction by A1;
  end;
  suppose j=7;
    then Sum digits(m,10) = 7 by A2,Th292;
    hence contradiction by A1;
  end;
  suppose j=8;
    then Sum digits(m,10) = 8 by A2,Th294;
    hence contradiction by A1;
  end;
  suppose j=9;
    then Sum digits(m,10) = 9 by A2,Th296;
    hence contradiction by A1;
  end;
  suppose j=10;
    then Sum digits(m,10) = 10 by A2,Th53;
    hence contradiction by A1;
  end;
  suppose j=11;
    then Sum digits(m,10) = 11 by A2,Th90;
    hence contradiction by A1;
  end;
  suppose j=12;
    then Sum digits(m,10) = 12 by A2,Th298;
    hence contradiction by A1;
  end;
  suppose j=13;
    then Sum digits(m,10) = 13 by A2,Th134;
    hence contradiction by A1;
  end;
  suppose j=14;
    then Sum digits(m,10) = 14 by A2,Th165;
    hence contradiction by A1;
  end;
  suppose j=15;
    then Sum digits(m,10) = 15 by A2,Th300;
    hence contradiction by A1;
  end;
  suppose j=16;
    then Sum digits(m,10) = 7 by A2,Th201;
    hence contradiction by A1;
  end;
  suppose j=17;
    then Sum digits(m,10) = 8 by A2,Th248;
    hence contradiction by A1;
  end;
  suppose j=18;
    then Sum digits(m,10) = 9 by A2,Th302;
    hence contradiction by A1;
  end;
  suppose j=19;
    then Sum digits(m,10) = 10 by A2,Th304;
    hence contradiction by A1;
  end;
  suppose j=20;
    then Sum digits(m,10) = 11 by A2,Th306;
    hence contradiction by A1;
  end;
  suppose j=21;
    then Sum digits(m,10) = 21 by A2,Th308;
    hence contradiction by A1;
  end;
  suppose j=22;
    then Sum digits(m,10) = 13 by A2,Th310;
    hence contradiction by A1;
  end;
  suppose j=23;
    then Sum digits(m,10) = 14 by A2,Th312;
    hence contradiction by A1;
  end;
  suppose j=24;
    then Sum digits(m,10) = 15 by A2,Th314;
    hence contradiction by A1;
  end;
  suppose j=25;
    then Sum digits(m,10) = 16 by A2,Th316;
    hence contradiction by A1;
  end;
  suppose j=26;
    then Sum digits(m,10) = 17 by A2,Th318;
    hence contradiction by A1;
  end;
  suppose j=27;
    then Sum digits(m,10) = 9 by A2,Th320;
    hence contradiction by A1;
  end;
  suppose j=28;
    then Sum digits(m,10) = 10 by A2,Th322;
    hence contradiction by A1;
  end;
  suppose j=29;
    then Sum digits(m,10) = 11 by A2,Th324;
    hence contradiction by A1;
  end;
  suppose j=30;
    then Sum digits(m,10) = 12 by A2,Th326;
    hence contradiction by A1;
  end;
  suppose j=31;
    then Sum digits(m,10) = 22 by A2,Th328;
    hence contradiction by A1;
  end;
  suppose j=32;
    then Sum digits(m,10) = 14 by A2,Th330;
    hence contradiction by A1;
  end;
  suppose j=33;
    then Sum digits(m,10) = 15 by A2,Th332;
    hence contradiction by A1;
  end;
  suppose j=34;
    then Sum digits(m,10) = 16 by A2,Th334;
    hence contradiction by A1;
  end;
  suppose j=35;
    then Sum digits(m,10) = 17 by A2,Th336;
    hence contradiction by A1;
  end;
  suppose j=36;
    then Sum digits(m,10) = 18 by A2,Th338;
    hence contradiction by A1;
  end;
  suppose j=37;
    then Sum digits(m,10) = 10 by A2,Th340;
    hence contradiction by A1;
  end;
  suppose j=38;
    then Sum digits(m,10) = 11 by A2,Th342;
    hence contradiction by A1;
  end;
  suppose j=39;
    then Sum digits(m,10) = 12 by A2,Th344;
    hence contradiction by A1;
  end;
  suppose j=40;
    then Sum digits(m,10) = 13 by A2,Th346;
    hence contradiction by A1;
  end;
  suppose j=41;
    then Sum digits(m,10) = 23 by A2,Th348;
    hence contradiction by A1;
  end;
  suppose j=42;
    then Sum digits(m,10) = 24 by A2,Th350;
    hence contradiction by A1;
  end;
  suppose j=43;
    then Sum digits(m,10) = 16 by A2,Th352;
    hence contradiction by A1;
  end;
  suppose j=44;
    then Sum digits(m,10) = 17 by A2,Th354;
    hence contradiction by A1;
  end;
  suppose j=45;
    then Sum digits(m,10) = 18 by A2,Th356;
    hence contradiction by A1;
  end;
end;
