The following presents a mathematical proof that the Galilean transformation equations are flawed, whenever either system is repositioned! The failure to inject from the repositioned system is the spawn for this mathematical failure. Wherein, the Galilean erroroneously, merely injects from the stationary system.

          
                                                                 injection of X into Y


Given Case 1 with a point P at 2 in A and a point P' at 2 in B with D = 0.
No depiction supplied D = 0; P with no P', as that would violate
( AXIOM 1 )      Coincident Cartesian coordinate systems specify equally aligned named points and coordinate points.
THEREFORE, case 1, is the only D = 0 condition to consider
If P = 2 in A, then P' = 2 in B.

case 2 and case 4 applied to case 1.
( AXIOM 2 )       Origin repositioning does not alter relative point coordination [ abscissal, ordinate, applicate ] as well as named points, with respect to the Cartesian coordinate system in which it resides.

Plan G(alilean) - Named points from the stationary system are injected into the repositioned system, whereas named points are not injected from the repositioned system into the stationary one. The author suggests that both are mandated mathematically to maintain mathematical equality ( Plan W ).

FINAL OBSERVATION ( an overview )

Regardless of how B is repositioned ( left or right wrt A ), x' in B is no longer 2 in B, when applying Plan G(alilean). Given coincident systems A and B, without a point P, then with repositioning of B, say to a distance of one hundred light years from A, then x' in B = 2. Both statements cannot be true. The truth; x in A = x' in B REGARDLESS of any displacement to either system related to the other.