(** Lemma 16.2 in dimension 2. *)

(** Lemma 16.12 in dimension 2. *)
Lemma characterization_of_congruence :
forall O E E' S U1 U2
A AX AY B BX BY C CX CY D DX DY
AXMBX AXMBX2 AYMBY AYMBY2 AB2
CXMDX CXMDX2 CYMDY CYMDY2 CD2,
Cd O E S U1 U2 A AX AY -> Cd O E S U1 U2 B BX BY ->
Cd O E S U1 U2 C CX CY -> Cd O E S U1 U2 D DX DY ->
Diff O E E' AX BX AXMBX -> Prod O E E' AXMBX AXMBX AXMBX2 ->
Diff O E E' AY BY AYMBY -> Prod O E E' AYMBY AYMBY AYMBY2 ->
Sum O E E' AXMBX2 AYMBY2 AB2 ->
Diff O E E' CX DX CXMDX -> Prod O E E' CXMDX CXMDX CXMDX2 ->
Diff O E E' CY DY CYMDY -> Prod O E E' CYMDY CYMDY CYMDY2 ->
Sum O E E' CXMDX2 CYMDY2 CD2 ->
(Cong A B C D <-> AB2 = CD2).