 Solution

For feasible points the duality gap decomposes asEach term in both sums is nonnegative. Therefore the gap is zero exactly when every product vanishes, namely the two sets of complementary slackness equations in the question (with $b-Ax$ or its negative making no difference to a zero product). If the equations hold, equality and weak duality prove optimality of both points. Conversely, if both are optimal and the optimal values are equal as assumed, the gap is zero, forcing each product to vanish. This proves both directions.