A group of people are at a party
The group is at a birthday party.
The fact that a group of people are at a party doesn't have to mean that they are at a birthday party.  They could be at any kind of party.
exists p1 x1 x2.(exists x3.(at(x1,x3) & n_party(x3)) & of(x1,x2) & n_people(x2) & n_group(x1))
exists p1 x1.(n_group(x1) & exists x2 x3.(at(x1,x2) & n_party(x2) & of(x2,x3) & n_birthday(x3)))