Uniqueness of Lagrangians in $T^{\ast }RP2$

We present a new and simpler proof of the fact that any Lagrangian $\mathbb{R}{P}^2$ in $T^{\ast }\mathbb{R}{P}^2$ is Hamiltonian isotopic to the zero section. Our proof mirrors the one given by Li and Wu for the Hamiltonian uniqueness of Lagrangians in $T^{\ast }{S}^2$, using surgery to turn Lagrangian spheres into symplectic ones. The main novel contribution is a detailed proof of the folklore fact that the complement of a symplectic quadric in $\mathbb{C}{P}^2$ can be identified with the unit cotangent disc bundle of $\mathbb{R}{P}^2$.