Let and be two cardinal numbers, then is, by definition, the cardinal of the disjoint union of two sets whose cardinalities are and respectively.
If in the definition of the cardinal addition the sets weren't assumed to be disjoint, that would not make sense: for instance we would have !
Therefore all you have to do is to prove the equality. By the way, this equlity holds even if (the case does not interest us for this problem, but we don't have to suppose and are disjoint to prove the equality)
The proof is quite simple, it can be done with two inclusions; the only things you have to use are the definitions of union and cartesian product of sets.