
theorem SDC:
  for a,b be Integer st a,b are_coprime holds (a - b)*(a + b) gcd a*b = 1
  proof
    let a,b be Integer such that
    A1: a,b are_coprime;
    a gcd b = a gcd (b - 1*a) & a gcd b = a gcd (b + 1*a) by NEWTON02:5; then
    -(a - b) gcd a = 1 & (a + b) gcd a = 1 by A1; then
    (a-b),a are_coprime & (a + b),a are_coprime by NEWTON02:1; then
    A2: (a - b)*(a + b),a are_coprime by INT_2:26;
    b gcd a = b gcd (a - 1*b) & b gcd a = b gcd (a + 1*b) by NEWTON02:5; then
    (a - b),b are_coprime & (a + b),b are_coprime by A1; then
    (a - b)*(a + b), b are_coprime by INT_2:26; then
    (a - b)*(a + b), a*b are_coprime by A2,INT_2:26;
    hence thesis;
  end;
