Is $T^2 \# \mathbb RP^2\cong \mathbb RP^2\# \mathbb RP^2\# \mathbb RP^2$?

Question

I was reading a little about how to imagine the projective plane and I have some weird intuition that says $T^2 \# \mathbb RP^2\cong \mathbb RP^2\# \mathbb RP^2\# \mathbb RP^2$. Is this true, and if so, is there any way to prove it formally using topology?

Added. If so, what is it? I've taken a first course on topology, so I know the basic definitions.