  The supercoonformal index on $\mathbb{RP}^{2} \times \mathbb{S}^{1}$ can be
derived exactly by the localization technique and applied to the direct proof
of Abelian mirror symmetry. We find two sets of parity conditions compatible
with the unorientable property of $\mathbb{RP}^{2}$ and then rigorously show
two kinds of Abelian mirror symmetry via the index on $\mathbb{RP}^{2} \times
\mathbb{S}^{1}$.
