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).