According to the proof method that two sets are equal, it is proved that both sides contain each other.

Let x∈f(A) exist, so x=f(z). Because z∈A, z∈X, z=g(f(z)).

Know x∈Y from x=f(z).

X=f(z)=f(g(f(z))=f(g(x))。

So x ∈ b.

So f(A) is included in b.

Take y∈B, then y∈Y, y=f(g(y)). Remember that u=g(y), then y=f(u). The following proves that u ∈ a.

Know u∈X from u=g(y).

And u=g(y)=g((f(u)).

So u∈A, so y=f(u)∈f(A).

So b is included in f(A).

So f (a) = b.