
theorem
  for a,b,c,d be positive Real st a + b = c + d holds
  max (a,b) = max (c,d) iff min (a,b) = min (c,d)
  proof
    let a,b,c,d be positive Real such that
    A1: a + b = c + d;
    ((max (a,b) = a & min (a,b) = b) or (max (a,b) = b & min (a,b) = a)) &
    ((max (c,d) = c & min (c,d) = d) or (max (c,d) = d & min (c,d) = c))
      by XXREAL_0:def 9,def 10;
    hence thesis by A1;
  end;
