The Klein bottle K^2 is a square where the opposite vertical edges are identified in the opposite direction and the horizontal edges are identified in the same direction.

Consider the space \mathbb{R}P^2 \# \mathbb{R}P^2 resulting from an annulus by identifying antipodal points on the outer circle, and also identifying antipodal points in the inner circle. Show that K^2 \cong \mathbb{R}P^2 \# \mathbb{R}P^2, i.e. K^2 is homeomorphic to \mathbb{R}P^2 \# \mathbb{R}P^2.