Definition proj1_conj (A B: Prop) (c: A /\ B): A :=
  match c with conj x _ => x end.Definition proj2_conj (A B: Prop) (c: A /\ B): B :=
  match c with conj _ x => x end.

Lemma eq_trans (X: Set) (a b c: X): a = b -> b = c -> a = c.