proposition: Lemma exists_grid_spec : exists S U1 U2, Cs PA PB S U1 U2.
proof: Proof. assert (~ Col PA PB PC) by (apply lower_dim). assert_diffs. destruct (ex_per_cong PB PA PA PC PA PB) as [J HJ]; Col; spliter. exists PA; exists PB; exists J. repeat (split; finish). Qed.
imports: GeoCoq.Tarski_dev.Ch15_lengths