Section PairPoset.
Variable (A B : poset).
Local Notation tp := (A * B)%type.
Definition pair_bot : tp := (bot, bot).
Definition pair_leq (p1 p2 : tp) := p1.1 <== p2.1 /\ p1.2 <== p2.2.

Lemma pair_trans x y z : pair_leq x y -> pair_leq y z -> pair_leq x z.