A rotation of angle theta around any point C is a function that maps point P to point P' such that CP = CP' and angle PCP' is theta (counterclockwise).
A reflection is the transformation that fixes every point on the line of reflection and associates to each point P not on that line a unique point P' such that the line of reflection is the perpendicular bisector of the line segment PP'.
I started working on this proof thinking that the point being rotated/reflected was on the given line m, but that doesn't have to be the case...