sorry for asking too many questions, but me and my friends are really lost in this class.. i was wondering if someone could help us on this question

prove theorem: Let (X,A_x) and (Y,A_y) be f_j(1≤j≤k) be (A_x,A_y) measurable functions, and g:R^k->R a continuous function. Then the functionϕ:X->R defined byϕ(x)=g(f_1(x),f_2(x),...,f_n(x)) is Borel-measurable.

prove corollary using the theorem: Let (X,A_x) be a measurable space, and let be f_1,f_2 be Borel-measurable functions on (X,A_x). Then the functions f_1+f_2 f_1*f_2 are also Borel-measurable on (X,A_x).

thank you!