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 py_to_pys : forall A B C, Py A B C = 1/2 * (Py A B A + Py B C B - Py A C A).