intros.
simpl in |- *.
red in |- *.
split; simpl in |- *; intro; red in H; red in |- *.
unfold transpose in |- *; destruct x; simpl in |- *.
simpl in H; intros; (apply Trans with (Ap2 b a); auto with algebra).
unfold transpose in H; destruct x; simpl in |- *.
simpl in H; intros.
auto with algebra.
Qed.
End defs.

(* Specifications: *)
Lemma symm_matr_subspace :
forall (n : Nat) (F : field), is_subspace (is_symmetric_pred F n).