Ltac construct_grid O U V H :=
elim grid_existence;let H:= fresh in (
intros O H; elim H; clear H;intros U H;elim H;clear H;intro V;intros H;decompose [and] H).

Lemma area_grid : forall O U V, perp O U O V ->
O ** U <> 0 -> O ** V <> 0 ->
2*2*(S O U V * S O U V) = O**U*O**U*O**V*O**V.