For mechanical systems with rheonomic couplings, it has been proved: 1. that n standard differential Lagrange equations of motion of the second kind (2.1) are not equivalent to the system of Lagrange equations of the first kind (1.4); for equivalence it is necessary to add equation (3.3) to equations (2.1). 2. On the basis of the n differential equations of Lagrange (2.1) or the corresponding 2n differential equations of Hamilton, it is impossible to prove the energy change theorem (1.1) or (1.6). To prove this theorem on the basis of equations (2.1), we must add equation (3.3) to them, i.e. justify the proof on the extended system of equations (4.1). 3. The relation of the 'theorem' or 'law' on energy change (5.2) is given in a formal way from equation (5.1) and does not express the energy change theorem corresponding to theorem (1.6). The invariant relation for the change in mechanical energy, which is equivalent to the relation (1.3) and (1.6) in the generalized independent Lagrage coordinates or relative to the canonical Hamiltonian coordinates, has the form (3.6) or (3.7), respectively.