Lemma characterization_of_collinearity : forall O E E' S U1 U2 A AX AY B BX BY C CX CY AXMBX AYMBY BXMCX BYMCY XProd YProd, 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 -> Diff O E E' AX BX AXMBX -> Diff O E E' AY BY AYMBY -> Diff O E E' BX CX BXMCX -> Diff O E E' BY CY BYMCY -> Prod O E E' AXMBX BYMCY XProd -> Prod O E E' AYMBY BXMCX YProd -> (Col A B C <-> XProd = YProd).