your proof of one is hard to follow.

we want to prove that S is 1-1, which means proving S(v) = 0 implies v = 0.

suppose S(v) = 0. since v is in V, and T is onto v, v = T(u) for some u in U.

therefore: S(v) = S(T(u)) = ST(u).

since S(v) = 0, ST(u) = 0.

since ST is 1-1, u = 0.

thus v = T(u) = T(0) = 0, which is what we desired to prove.

2) here, we must find for any given v in V, some u in U with T(u) = v.

now, we know that given any w in W, we have u in U, with ST(u) = w.

so consider w = S(v), for our given v above.

we have w = S(v) = ST(u), for some u in U (since ST is onto).

since S is 1-1, and S(v) = ST(u) = S(T(u)), v = T(u), and we are done.