
theorem
  for a,b,c be Nat st a|^2 + b|^2 = c|^2 holds
    b|^2 mod a = c|^2 mod a & a|^2 mod b = c|^2 mod b
  proof
    let a,b,c be Nat such that
    A1: a|^2 + b|^2 = c|^2;
    A2: (a|^2 + b|^2) mod a = ((a|^2 mod a) + (b|^2 mod a)) mod a by NAT_D:66
    .= b|^2 mod a;
    (a|^2 + b|^2) mod b = ((a|^2 mod b) + (b|^2 mod b)) mod b by NAT_D:66
    .= a|^2 mod b;
    hence thesis by A1,A2;
  end;
