I've been reading the Quantum Mechanics for Mathematicians book by Leon A. Takhtajan. On p. 15 (which is p. 29 in the PDF file), he introduces the energy $E = \sum_i \dot q^i\frac{\partial L}{\partial \dot q^i} - L$ of a Lagrangian system and claims that $E$ is well defined as a function from $TM \times \mathbb{R}$, where $M$ is the configuration space, $TM$ is the tangent bundle, and $\mathbb R$ the time axis. To "prove" this, he takes two arbitrary charts on $M$ (as opposed to $TM$) and the corresponding coordinate transformation $\mathbf{q'} = F(\mathbf{q})$. He extends $F$ to a coordinate transformation on $TM$ using the tangent map $F_*$ and concludes that the energy $E$ is invariant under the extended transformation, and so the function $E: TM \times \mathbb{R} \to \mathbb{R}$ is well defined since the result does not depend on the coordinate representation of the argument.
Problem is: THIS IS NOT A PROOF, as it only proves the claim for one special case. Should we be starting with an arbitrary coordinate transformation on $TM \times \mathbb{R}$ instead?
I tried to find an answer online but couldn't. So, my questions are:
Is $E$ really well defined as a function on $TM \times \mathbb{R}$, as the book claims?
Is there an online source that gives the correct proof of this fact?
To simplify the question: I am aware that the Lagrangian $L: TM\times \mathbb{R} \to \mathbb{R}$ is well defined, so it's the sum $\sum_i \dot q^i \frac{\partial L}{\partial \dot q^i}$ that worries me.