
theorem
Sierp36 209,11
proof
  thus Sum digits(209,10) = 11 by Th90;
  209=19*11;
  hence 11 divides 209 by INT_1:def 3;
  let m be Nat;
  assume A1: Sum digits(m,10) = 11 & 11 divides m;
  then consider j being Nat such that
  A2: m=11*j by NAT_D:def 3;
  assume m < 209;
  then 11*j < 11*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) = 2 by A2,Th56;
    hence contradiction by A1;
  end;
  suppose j=2;
    then Sum digits(m,10) = 4 by A2,Th58;
    hence contradiction by A1;
  end;
  suppose j=3;
    then Sum digits(m,10) = 6 by A2,Th60;
    hence contradiction by A1;
  end;
  suppose j=4;
    then Sum digits(m,10) = 8 by A2,Th62;
    hence contradiction by A1;
  end;
  suppose j=5;
    then Sum digits(m,10) = 10 by A2,Th64;
    hence contradiction by A1;
  end;
  suppose j=6;
    then Sum digits(m,10) = 12 by A2,Th66;
    hence contradiction by A1;
  end;
  suppose j=7;
    then Sum digits(m,10) = 14 by A2,Th68;
    hence contradiction by A1;
  end;
  suppose j=8;
    then Sum digits(m,10) = 16 by A2,Th70;
    hence contradiction by A1;
  end;
  suppose j=9;
    then Sum digits(m,10) = 18 by A2,Th72;
    hence contradiction by A1;
  end;
  suppose j=10;
    then Sum digits(m,10) = 2 by A2,Th37;
    hence contradiction by A1;
  end;
  suppose j=11;
    then Sum digits(m,10) = 4 by A2,Th74;
    hence contradiction by A1;
  end;
  suppose j=12;
    then Sum digits(m,10) = 6 by A2,Th76;
    hence contradiction by A1;
  end;
  suppose j=13;
    then Sum digits(m,10) = 8 by A2,Th78;
    hence contradiction by A1;
  end;
  suppose j=14;
    then Sum digits(m,10) = 10 by A2,Th80;
    hence contradiction by A1;
  end;
  suppose j=15;
    then Sum digits(m,10) = 12 by A2,Th82;
    hence contradiction by A1;
  end;
  suppose j=16;
    then Sum digits(m,10) = 14 by A2,Th84;
    hence contradiction by A1;
  end;
  suppose j=17;
    then Sum digits(m,10) = 16 by A2,Th86;
    hence contradiction by A1;
  end;
  suppose j=18;
    then Sum digits(m,10) = 18 by A2,Th88;
    hence contradiction by A1;
  end;
end;
