Encoding Theory – Do you prove that the MinAveCodeLen of a product information source is smaller than the sum of the multiplicand and multiplier sources?

The product of 2 independent sources $ (S_A, P_A) $ and $ (S_B, P_B) $ is defined as

$$
(S, P) text {s.t. } S = {s_As_B | s_A in S_A, s_B in B } text {and} P (s_As_B) = P_A (s_A) cdot P_B (s_B) , forall s_A in S_A, s_B in S_B
$$

and MinAveCodeLen is the minimum average codeword length corresponding to the average codeword length of a Huffman coding.

The problem is to prove

$$
mathrm {MinAveCodeLen} (P) leq mathrm {MinAveCodeLen} (P_A) + mathrm {MinAveCodeLen} (P_B)
$$

The original problem is quite confusing. I am asked to prove $ H (P_A) + H (P_B) = H (P) $ in the previous part I tried to connect the problem with it $ H $ initially without progress.