I know this probably has a really straightforward answer, specially if it is as standard as it is entuitive to visualize. Still, because i'm not experienced at all on working with this objects and im just learning by myself, i would like to see a formal proof of the following facts, please : (# means the CONNECTED SUM) (i) $(\mathbb{R}\mathbb{P}^{2})\#(\mathbb{R} \mathbb{P}^{2}) = K^{2}$ (where $\mathbb{R}\mathbb{P}^{2}$ is the projective plane and $K^{2}$ is the Klein Bottle) (ii) $(\mathbb{R}\mathbb{P}^{2})\#(T^{2}) = (\mathbb{R}\mathbb{P}^{2})\#(K^{2})$ (where $T^{2}$ is a torus)
Im sorry if the question is annoying, but i dont know how to start, i can just see how it works if i draw or visualize, but i would like to see a rigourous proof. Again, sorry if its really obvious, but im just learning this by myself.
Thank you a lot