If $a^k\equiv b\bmod p$ then $a^{k+1}\equiv ab\bmod p$.

Question

If $a^k\equiv b\bmod p$ then $a^{k+1}\equiv ab\bmod p$.

Prove using the proposition, If $a\equiv b\bmod n$ and $c\equiv d\bmod n$ then $ac\equiv bd\bmod n$.