
theorem
  for a,b,c,d be Nat st a,b are_coprime holds
    a*b = c*d implies a*b = (a gcd c)*(b gcd c)*(a gcd d)*(b gcd d)
  proof
    let a,b,c,d be Nat such that
    A1: a,b are_coprime;
    assume
    A2: a*b = c*d; then
    c divides a*b & d divides a*b; then
    (a gcd c)*(b gcd c) = c & (a gcd d)*(b gcd d) = d by A1,CDN;
    hence thesis by A2;
  end;
