
theorem
Sierp36 247,13
proof
  thus Sum digits(247,10) = 13 by Th134;
  247=19*13;
  hence 13 divides 247 by INT_1:def 3;
  let m be Nat;
  assume A1: Sum digits(m,10) = 13 & 13 divides m;
  then consider j being Nat such that
  A2: m=13*j by NAT_D:def 3;
  assume m < 247;
  then 13*j < 13*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) = 4 by A2,Th102;
    hence contradiction by A1;
  end;
  suppose j=2;
    then Sum digits(m,10) = 8 by A2,Th104;
    hence contradiction by A1;
  end;
  suppose j=3;
    then Sum digits(m,10) = 12 by A2,Th106;
    hence contradiction by A1;
  end;
  suppose j=4;
    then Sum digits(m,10) = 7 by A2,Th108;
    hence contradiction by A1;
  end;
  suppose j=5;
    then Sum digits(m,10) = 11 by A2,Th110;
    hence contradiction by A1;
  end;
  suppose j=6;
    then Sum digits(m,10) = 15 by A2,Th112;
    hence contradiction by A1;
  end;
  suppose j=7;
    then Sum digits(m,10) = 10 by A2,Th114;
    hence contradiction by A1;
  end;
  suppose j=8;
    then Sum digits(m,10) = 5 by A2,Th116;
    hence contradiction by A1;
  end;
  suppose j=9;
    then Sum digits(m,10) = 9 by A2,Th118;
    hence contradiction by A1;
  end;
  suppose j=10;
    then Sum digits(m,10) = 4 by A2,Th41;
    hence contradiction by A1;
  end;
  suppose j=11;
    then Sum digits(m,10) = 8 by A2,Th78;
    hence contradiction by A1;
  end;
  suppose j=12;
    then Sum digits(m,10) = 12 by A2,Th120;
    hence contradiction by A1;
  end;
  suppose j=13;
    then Sum digits(m,10) = 16 by A2,Th122;
    hence contradiction by A1;
  end;
  suppose j=14;
    then Sum digits(m,10) = 11 by A2,Th124;
    hence contradiction by A1;
  end;
  suppose j=15;
    then Sum digits(m,10) = 15 by A2,Th126;
    hence contradiction by A1;
  end;
  suppose j=16;
    then Sum digits(m,10) = 10 by A2,Th128;
    hence contradiction by A1;
  end;
  suppose j=17;
    then Sum digits(m,10) = 5 by A2,Th130;
    hence contradiction by A1;
  end;
  suppose j=18;
    then Sum digits(m,10) = 9 by A2,Th132;
    hence contradiction by A1;
  end;
end;
