
theorem COM:
  for a,b be Nat st a,b are_coprime holds
    for n be non trivial Nat holds max (a mod n, b mod n) > 0
  proof
    let a,b be Nat such that
    A1: a,b are_coprime;
    assume not thesis; then
    consider m be non trivial Nat such that
    A2: max (a mod m, b mod m) <= 0;
    a mod m = 0 & b mod m = 0 by A2; then
    m divides a & m divides b by INT_1:62;
    hence contradiction by A1,PYTHTRIP:def 1,NAT_2:def 1;
  end;
