Section trivial_lemmas.
simpl in |- *.
red in |- *.
simpl in |- *.
intros; destruct i; destruct j.
simpl in |- *; simpl in H.
apply Kronecker_case_unequal; auto with algebra.
Qed.
intros; red in H, H0.
red in |- *.
intros.
case (nat_total_order _ _ H1); intro; auto with algebra.
Qed.
End trivial_lemmas.

Lemma up_triang_subspace :
forall (F : field) (m n : Nat), is_subspace (is_upper_triangular_pred F m n).