I propose a Lagrangian proof of Einstein's well-known law that the mass system is its internal energy. The interest of this proof is to show how the distinction between internal degrees of freedom and the center of mass appears in the Lagrangian formalism. Considering that the Lagrangian depends on a particular set of variables for the internal degree of freedom, I show in a standard Lagrangian way how one can naturally find the desired law. This proof does not use the tensors of energy-momentum and can be easily used by students familiar with Lagrangian mechanics and the basis of special relativity. I apply the method for the particles and for the field, using the scalar field for simplification but it is easy to generalize for other fields (containing only the first derivative in Lagrangian). I give the example for the gravitation field. The method permits us to observe a strong relation between the Einstein’s E=mc² law and his other famous law of the time dilatation. I carefully analyze the meaning of the particular choice of the variable and showing a sort of a modified speed addition formula without contradicting, of course, the one of Einstein (& Poincaré). I also try to untangle (for myself at least) the relation between the mass and the origin of the energy scale. Finally I analyze the reason why in Newtonian mechanic we don’t have a such law. In future complement I will apply this way of thinking in the toy model of the electron (useful for an explicit classical renormalization of the mass) and the effective description of a complex system in term of a particle in order to better understand the passage from this 2 forms of description often used but never really explained.