
theorem
Sierp36 399,21
proof
  thus Sum digits(399,10) = 21 by Th308;
  399=19*21;
  hence 21 divides 399 by INT_1:def 3;
  let m be Nat;
  assume A1: Sum digits(m,10) = 21 & 21 divides m;
  then consider j being Nat such that
  A2: m=21*j by NAT_D:def 3;
  assume m < 399;
  then 21*j < 21*19 by A2;
  then j < 18+1 by XREAL_1:64;
  then j <= 18 by NAT_1:9;
  then j=0 or ... or j=18;
  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) = 3 by A2,Th730;
    hence contradiction by A1;
  end;
  suppose j=2;
    then Sum digits(m,10) = 6 by A2,Th141;
    hence contradiction by A1;
  end;
  suppose j=3;
    then Sum digits(m,10) = 9 by A2,Th732;
    hence contradiction by A1;
  end;
  suppose j=4;
    then Sum digits(m,10) = 12 by A2,Th145;
    hence contradiction by A1;
  end;
  suppose j=5;
    then Sum digits(m,10) = 6 by A2,Th174;
    hence contradiction by A1;
  end;
  suppose j=6;
    then Sum digits(m,10) = 9 by A2,Th151;
    hence contradiction by A1;
  end;
  suppose j=7;
    then Sum digits(m,10) = 12 by A2,Th734;
    hence contradiction by A1;
  end;
  suppose j=8;
    then Sum digits(m,10) = 15 by A2,Th153;
    hence contradiction by A1;
  end;
  suppose j=9;
    then Sum digits(m,10) = 18 by A2,Th736;
    hence contradiction by A1;
  end;
  suppose j=10;
    then Sum digits(m,10) = 3 by A2,Th157;
    hence contradiction by A1;
  end;
  suppose j=11;
    then Sum digits(m,10) = 6 by A2,Th738;
    hence contradiction by A1;
  end;
  suppose j=12;
    then Sum digits(m,10) = 9 by A2,Th163;
    hence contradiction by A1;
  end;
  suppose j=13;
    then Sum digits(m,10) = 12 by A2,Th740;
    hence contradiction by A1;
  end;
  suppose j=14;
    then Sum digits(m,10) = 15 by A2,Th742;
    hence contradiction by A1;
  end;
  suppose j=15;
    then Sum digits(m,10) = 9 by A2,Th744;
    hence contradiction by A1;
  end;
  suppose j=16;
    then Sum digits(m,10) = 12 by A2,Th205;
    hence contradiction by A1;
  end;
  suppose j=17;
    then Sum digits(m,10) = 15 by A2,Th252;
    hence contradiction by A1;
  end;
  suppose j=18;
    then Sum digits(m,10) = 18 by A2,Th746;
    hence contradiction by A1;
  end;
end;
