
theorem
  for a,b,c,d be positive Real st a + b = c + d holds
    1/a + 1/b > 1/c + 1/d iff a*b < c*d
  proof
    let a,b,c,d be positive Real such that
    A1: a + b = c + d;
    (1/a*a)*b = 1*b & (1/b*b)*a = 1*a & (1/c*c)*d = 1*d & (1/d*d)*c = 1*c
      by XCMPLX_1:87; then
    a + b = (1/a + 1/b)*(a*b) & c + d = (1/c + 1/d)*(c*d);
    hence thesis by A1,XREAL_1:98;
  end;
