
theorem
  for a,b be non zero Nat st a|^2 - b|^2 in NAT holds
    (a|^2 - b|^2) mod (2*b) = (a|^2 + b|^2) mod (2*b)
  proof
    let a,b be non zero Nat such that
    A1: a|^2 - b|^2 in NAT;
    A2: b|^2 = b*b by NEWTON:81;
    (a|^2 - b|^2) mod 2*b|^2 = (a|^2 + b|^2) mod 2*b|^2 by MRS; then
    (a|^2 - b|^2) mod ((2*b)*b) = (a|^2 + b|^2) mod ((2*b)*b) by A2;
    hence thesis by A1,RMI;
  end;
