Prove that every maximal flow yields the same minimal cut
Here are some hints.  Instead of trying to prove the contrapositive, try proving the statement directly.  If $f_1 \ne f_2$, then that means that there is some edge $e$ where $f_1$ sends a different amount of flow down than $f_2$ does.  What does that imply about the residual graph for $f_1$, vs the residual graph for $f_2$?  Those two residual graphs must differ in some specific way: how?  What does that difference imply for $S_1$ vs $S_2$?  Try writing out some small examples, with small graphs, where you draw the flows, residual graphs, and cuts explicitly.  What do you find?

I suggest you spend a bit more quality time on your own, trying a bit harder to solve it yourself.  This is your exercise; you need to solve it yourself.  And you need to put in a bit more effort to explore this problem.  Don't give up easily -- you can do it!