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So let x [itex]\in[/itex] f^-1(f(E)) => thus f(x) [itex]\in[/itex] f(E) => x[itex]\in[/itex] E

So I have proved that x is a point within E, a subset of A, to me I think I am missing something and have not proved f^-1(f(E)) = E.

any suggestions?