
theorem
Sierp36 448,16
proof
  thus Sum digits(448,10) = 16 by Th219;
  448=28*16;
  hence 16 divides 448 by INT_1:def 3;
  let m be Nat;
  assume A1: Sum digits(m,10) = 16 & 16 divides m;
  then consider j being Nat such that
  A2: m=16*j by NAT_D:def 3;
  assume m < 448;
  then 16*j < 16*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) = 7 by A2,Th179;
    hence contradiction by A1;
  end;
  suppose j=2;
    then Sum digits(m,10) = 5 by A2,Th181;
    hence contradiction by A1;
  end;
  suppose j=3;
    then Sum digits(m,10) = 12 by A2,Th99;
    hence contradiction by A1;
  end;
  suppose j=4;
    then Sum digits(m,10) = 10 by A2,Th183;
    hence contradiction by A1;
  end;
  suppose j=5;
    then Sum digits(m,10) = 8 by A2,Th31;
    hence contradiction by A1;
  end;
  suppose j=6;
    then Sum digits(m,10) = 15 by A2,Th185;
    hence contradiction by A1;
  end;
  suppose j=7;
    then Sum digits(m,10) = 4 by A2,Th149;
    hence contradiction by A1;
  end;
  suppose j=8;
    then Sum digits(m,10) = 11 by A2,Th187;
    hence contradiction by A1;
  end;
  suppose j=9;
    then Sum digits(m,10) = 9 by A2,Th189;
    hence contradiction by A1;
  end;
  suppose j=10;
    then Sum digits(m,10) = 7 by A2,Th47;
    hence contradiction by A1;
  end;
  suppose j=11;
    then Sum digits(m,10) = 14 by A2,Th84;
    hence contradiction by A1;
  end;
  suppose j=12;
    then Sum digits(m,10) = 12 by A2,Th191;
    hence contradiction by A1;
  end;
  suppose j=13;
    then Sum digits(m,10) = 10 by A2,Th128;
    hence contradiction by A1;
  end;
  suppose j=14;
    then Sum digits(m,10) = 8 by A2,Th159;
    hence contradiction by A1;
  end;
  suppose j=15;
    then Sum digits(m,10) = 6 by A2,Th193;
    hence contradiction by A1;
  end;
  suppose j=16;
    then Sum digits(m,10) = 13 by A2,Th195;
    hence contradiction by A1;
  end;
  suppose j=17;
    then Sum digits(m,10) = 11 by A2,Th197;
    hence contradiction by A1;
  end;
  suppose j=18;
    then Sum digits(m,10) = 18 by A2,Th199;
    hence contradiction by A1;
  end;
  suppose j=19;
    then Sum digits(m,10) = 7 by A2,Th201;
    hence contradiction by A1;
  end;
  suppose j=20;
    then Sum digits(m,10) = 5 by A2,Th203;
    hence contradiction by A1;
  end;
  suppose j=21;
    then Sum digits(m,10) = 12 by A2,Th205;
    hence contradiction by A1;
  end;
  suppose j=22;
    then Sum digits(m,10) = 10 by A2,Th207;
    hence contradiction by A1;
  end;
  suppose j=23;
    then Sum digits(m,10) = 17 by A2,Th209;
    hence contradiction by A1;
  end;
  suppose j=24;
    then Sum digits(m,10) = 15 by A2,Th211;
    hence contradiction by A1;
  end;
  suppose j=25;
    then Sum digits(m,10) = 4 by A2,Th213;
    hence contradiction by A1;
  end;
  suppose j=26;
    then Sum digits(m,10) = 11 by A2,Th215;
    hence contradiction by A1;
  end;
  suppose j=27;
    then Sum digits(m,10) = 9 by A2,Th217;
    hence contradiction by A1;
  end;
end;
