  Let A be a based subspace of Y. Under the assumptions that Y is
path-connected and that the reduced diagonal map of A induces the zero map in
all mod 2 reduced homology groups, we compute a formula for the mod 2 reduced
Poincar\'{e} series of the loop space $\Omega ((A \wedge \mathbb{RP}^\infty)
\cup_{A \wedge \mathbb{RP}^1} (Y \wedge \mathbb{RP}^1))$. Here
$\mathbb{RP}^\infty$ and $\mathbb{RP}^1$ denote the infinite real projective
space and the real projective line respectively.
