Answer to Question #37613 in Geometry for amal

Question

Hi
Recall that we can thought of a projective plane as an upper hemisphere. We define the binary operation on a projective plane as follows: Let x and y be any two points belong to the projective plane, think of the projective plane as an upper hemisphere. Then take the antipodal point of y which is in the lower hemispher and identified with y. Draw a line through x and the antipodal point of y. Then draw a line parallel to that line with same length and the origin is the mid point, x*y is then defined as the endpoint of this line (Since the end points are antipodal so they are equal in RP).
My question is
Is it true that x*(y*z)=(x*y)*(x*z) for x,y,z in RP
If yes, then can you prove it for me please

thank you in advance