
theorem
  for a,b be positive Real holds a + b < a * b iff 1/a + 1/b < 1
  proof
    let a,b be positive Real;
    thus a + b < a * b implies 1/a + 1/b < 1
    proof
      a + b < a*b implies 1/a + 1/b <= 1 & 1/a + 1/b <> 1 by SIO,SIL;
      hence thesis by XXREAL_0:1;
    end;
    1/a + 1/b < 1 implies a + b <= a * b & a + b <> a * b by SIO,SIL;
    hence thesis by XXREAL_0:1;
  end;
