Equaivalent Statements?

[loop quantum gravity]

Is the statement: the function f:S^n->S^n (the n unit sphere), f(x)=-x preserves orientation iff n is odd
equivalent to: PR^n is orientable iff n is odd where PR^n is the projective real space.

I think the answer is yes, just want to be sure here.

thanks in advance.

[gel]

yes.