  In this paper, we establish first the resonance identity for non-contractible
homologically visible prime closed geodesics on Finsler $n$-dimensional real
projective space $(\mathbb{R}P^n,F)$ when there exist only finitely many
distinct non-contractible closed geodesics on $(\mathbb{R}P^n,F)$, where the
integer $n\geq2$. Then as an application of this resonance identity, we prove
the existence of at least two distinct non-contractible closed geodesics on
$\mathbb{R}P^{n}$ with a bumpy and irreversible Finsler metric. Together with
two previous results on bumpy and reversible Finsler metrics in \cite{DLX2015}
and \cite{Tai2016}, it yields that every $\mathbb{R}P^{n}$ with a bumpy Finsler
metric possesses at least two distinct non-contractible closed geodesics.
