Record bijection (A B:Type) : Type := Bijection {
  bij_f: A -> B;
  bij_g: B -> A;
  bij_fg: forall x, bij_f (bij_g x) = x;
  bij_gf: forall x, bij_g (bij_f x) = x
}.Record bijection (A B:Type) : Type := Bijection {
  bij_f: A -> B;
  bij_g: B -> A;
  bij_fg: forall x, bij_f (bij_g x) = x;
  bij_gf: forall x, bij_g (bij_f x) = x
}.Record bijection (A B:Type) : Type := Bijection {
  bij_f: A -> B;
  bij_g: B -> A;
  bij_fg: forall x, bij_f (bij_g x) = x;
  bij_gf: forall x, bij_g (bij_f x) = x
}.Record bijection (A B:Type) : Type := Bijection {
  bij_f: A -> B;
  bij_g: B -> A;
  bij_fg: forall x, bij_f (bij_g x) = x;
  bij_gf: forall x, bij_g (bij_f x) = x
}.Record bijection (A B:Type) : Type := Bijection {
  bij_f: A -> B;
  bij_g: B -> A;
  bij_fg: forall x, bij_f (bij_g x) = x;
  bij_gf: forall x, bij_g (bij_f x) = x
}.Record bijection (A B:Type) : Type := Bijection {
  bij_f: A -> B;
  bij_g: B -> A;
  bij_fg: forall x, bij_f (bij_g x) = x;
  bij_gf: forall x, bij_g (bij_f x) = x
}.

Lemma bij_gf_id: forall {A B} (f: bijection A B), (bij_g _ _ f) oo (bij_f _ _ f) = id _.