
theorem N02193:
  for a,b be square Nat st a,b are_coprime holds not 3 divides a + b
  proof
    let a,b be square Nat such that
    A1: a,b are_coprime;
    consider x be Nat such that
    A2: a = x^2 by PYTHTRIP:def 3;
    consider y be Nat such that
    A3: b = y^2 by PYTHTRIP:def 3;
    A4: x^2 = x|^2 & y^2 = y|^2 by NEWTON:81;
    1 = 2-root (1|^2)
    .= 2-root ((x gcd y)|^2) by A1,A2,A3,A4,NEWTON03:4
    .= x gcd y; then
    x,y are_coprime;
    hence thesis by A2,A3,A4,NEWTON02:193;
  end;
