Structural induction proof
Here is the complete proof:

reverse(append(Cons(w,ws),z)) =
reverse(Cons(w,append(ws,z))) =
append(reverse(append(ws,z)),w) =
append(Cons(z,reverse(ws)),w) =
Cons(z,append(reverse(ws),w)) =
Cons(z,reverse(Cons(w,ws)))

At one point we used the inductive hypothesis.