Could someone please check my work

.WS. Prove: Let x, y ( R then (xyl = lac| . | y] . Suppose x,y E R. case one: Let x _ 0 and y _ 0. Then x| . ly| = xy = lay by definition 3.2.9 for the absolute value. case two: Let x 0. Then 12 . ly = (- 2) . (y) = - xy = lay by definition 3.2.9 for the absolute value. In any case, lac| . ly) = [acyl. Therefore ifx, y E R , then lacy) = [ac] . | y]