Hyper-triangular numbers

Triangular number $T_n$ is defined as $T_n = 1 + 2 + \dots + n$.

Tetrahedral number $R_n$ is defined as $R_n = T_1 + T_2 + \dots + T_n$.

Let $T^k_n$ denote the $k$-th dimensional triangular numbers, with $T^1_n = 1+1+\dots+1 = n$, $T^2_n = T_n$, and $T^3_n = R_n$. Specifically, the higher dimensional triangular numbers are defined as: $$ T^{k+1}_n = T^k_1 + T^k_2 + \dots + T^k_n$$ Prove that: $$T^k_n.T^m_1 + T^k_{n-1}.T^m_2 + \dots + T^k_1.T^m_n = T^{k+m+1}_{n+1}$$