Tangent Bundle of Product Manifold

Question

Suppose $M,N$ are manifolds, and consider the product $M\times N$. From this answer, I know that:

$T_{(m,n)}(M \times N) \cong T_m M \oplus T_n N $

Can we conclude that $T(M\times N) \cong T(M) \oplus T(N)$