Ok.

So,

Let $\displaystyle

\,h, h', e\in H\,,\,k, k', e'\in K\$

(hh',k) = hh' (def

)

= h $\displaystyle x$ h'

=

(h,k) $\displaystyle x$

(h',k) (def operation H x K)

Since

(hh',k) =

(h,k) $\displaystyle x$

(h',k) is a homomorphism

Since

a homomorphism let $\displaystyle \,e\in H\,,\,e'\in K\$ be the identities.

(h,e) = h (def

)

Therefore,

a surjective homomorphism.

By a similar argument, $\displaystyle \pi_2$

is also a surjective homomorphism.
QED

This feels very wrong to me...but rarely do proofs ever feel right.

Am I on the right track?