 But then Pu's systolic inequality asserts precisely that a Riemannian projective plane of given systole has minimum area if and only if it is round (that is, obtained from a Euclidean sphere by identifying each point with its opposite).The area of this round projective plane equals the area of the hemisphere (because each of them has half the area of the sphere).The proof of Pu's inequality relies, in turn, on the uniformization theorem.