Regular functions on distinguished open sets

Question

I’ve been reviewing algebraic geometry by reading through A. Gathmann’s notes. Somehow I don’t understand few steps included in the following proof. I appreciate any help :) \

1. How does $V(f_a)=V(h_a)$ imply that we can assume these two functions are the same?\
2. Why are these two functions $$g_af_b=g_bf_a$$ both zero as he claims in the paragraph below the equation (*)?
   As for the first question, I understand that if their difference is in the ideal $I(X)$, then they can be used interchangeably. However, is it the case here? We do know that their vanishing sets are the same, but we are not so sure about how they behave on the distinguished set.