
theorem
Sierp36 476,17
proof
  thus Sum digits(476,10) = 17 by Th266;
  476=28*17;
  hence 17 divides 476 by INT_1:def 3;
  let m be Nat;
  assume A1: Sum digits(m,10) = 17 & 17 divides m;
  then consider j being Nat such that
  A2: m=17*j by NAT_D:def 3;
  assume m < 476;
  then 17*j < 17*28 by A2;
  then j < 27+1 by XREAL_1:64;
  then j <= 27 by NAT_1:9;
  then j=0 or ... or j=27;
  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) = 8 by A2,Th222;
    hence contradiction by A1;
  end;
  suppose j=2;
    then Sum digits(m,10) = 7 by A2,Th224;
    hence contradiction by A1;
  end;
  suppose j=3;
    then Sum digits(m,10) = 6 by A2,Th226;
    hence contradiction by A1;
  end;
  suppose j=4;
    then Sum digits(m,10) = 14 by A2,Th228;
    hence contradiction by A1;
  end;
  suppose j=5;
    then Sum digits(m,10) = 13 by A2,Th230;
    hence contradiction by A1;
  end;
  suppose j=6;
    then Sum digits(m,10) = 3 by A2,Th232;
    hence contradiction by A1;
  end;
  suppose j=7;
    then Sum digits(m,10) = 11 by A2,Th234;
    hence contradiction by A1;
  end;
  suppose j=8;
    then Sum digits(m,10) = 10 by A2,Th236;
    hence contradiction by A1;
  end;
  suppose j=9;
    then Sum digits(m,10) = 9 by A2,Th238;
    hence contradiction by A1;
  end;
  suppose j=10;
    then Sum digits(m,10) = 8 by A2,Th49;
    hence contradiction by A1;
  end;
  suppose j=11;
    then Sum digits(m,10) = 16 by A2,Th86;
    hence contradiction by A1;
  end;
  suppose j=12;
    then Sum digits(m,10) = 6 by A2,Th240;
    hence contradiction by A1;
  end;
  suppose j=13;
    then Sum digits(m,10) = 5 by A2,Th130;
    hence contradiction by A1;
  end;
  suppose j=14;
    then Sum digits(m,10) = 13 by A2,Th161;
    hence contradiction by A1;
  end;
  suppose j=15;
    then Sum digits(m,10) = 12 by A2,Th242;
    hence contradiction by A1;
  end;
  suppose j=16;
    then Sum digits(m,10) = 11 by A2,Th197;
    hence contradiction by A1;
  end;
  suppose j=17;
    then Sum digits(m,10) = 19 by A2,Th244;
    hence contradiction by A1;
  end;
  suppose j=18;
    then Sum digits(m,10) = 9 by A2,Th246;
    hence contradiction by A1;
  end;
  suppose j=19;
    then Sum digits(m,10) = 8 by A2,Th248;
    hence contradiction by A1;
  end;
  suppose j=20;
    then Sum digits(m,10) = 7 by A2,Th250;
    hence contradiction by A1;
  end;
  suppose j=21;
    then Sum digits(m,10) = 15 by A2,Th252;
    hence contradiction by A1;
  end;
  suppose j=22;
    then Sum digits(m,10) = 14 by A2,Th254;
    hence contradiction by A1;
  end;
  suppose j=23;
    then Sum digits(m,10) = 13 by A2,Th256;
    hence contradiction by A1;
  end;
  suppose j=24;
    then Sum digits(m,10) = 12 by A2,Th258;
    hence contradiction by A1;
  end;
  suppose j=25;
    then Sum digits(m,10) = 11 by A2,Th260;
    hence contradiction by A1;
  end;
  suppose j=26;
    then Sum digits(m,10) = 10 by A2,Th262;
    hence contradiction by A1;
  end;
  suppose j=27;
    then Sum digits(m,10) = 18 by A2,Th264;
    hence contradiction by A1;
  end;
end;
