Let $Q_1$ be an $m×n$ matrix with orthonormal columns and $Q_2$ be an $n×p$ matrix with orthonormal columns

Question

Let $Q_1$ be an $m×n$ matrix with orthonormal columns and $Q_2$ be an $n×p$ matrix with orthonormal columns. Show that $Q_1Q_2$ has orthonormal columns.

My proof.

Check if $Q^TQ = I$ for $Q_1Q_2$, check $(Q_1Q_2)^TQ_1Q_2 = Q_2^TQ_1^TQ_1Q_2 = Q_t^TIQ_2 = I$ so therefor I have proved my question