
theorem
  for a,b,c,d be positive Real holds
  max (a,b) = max (c,d) & min (a,b) = min (c,d) iff (a*b = c*d & a+b = c+d)
  proof
    let a,b,c,d be positive Real;
    A1: a + b = max (a,b) + min (a,b) & c + d = max (c,d) + min (c,d) &
    a * b = max (a,b)*min(a,b) & max (c,d)*min(c,d) = c * d by NEWTON04:18;
    a * b = c * d & a + b = c + d implies max (a,b) = max (c,d) &
      min (a,b) = min (c,d)
    proof
      assume
      B1: a * b = c * d & a + b = c + d; then
      max (a,b) = max (c,d) by ISM;
      hence thesis by A1,B1;
    end;
    hence thesis by A1;
  end;
