*Proof*.
Let

be any element in

. As

is a surjection, there exists at least one element

in

such that

. As

is a surjection, this element

has at least one pre-image

, and this element is a pre-image of

by

. This shows that every element of

has at least one pre-image in

by

, i.e. that

is surjective.

*Proof*.
Let

be any element in

. As

is surjective, there exists at least one element

such that

. Thus

is a pre-image of

by

. This shows that every element of

has a pre-image by

, i.e.

is surjective.