I have the following definitions:

* A transformation is said to be **quasistatic** if it passes only through equilibrium states (that is, the thermodynamical coordinates of the system are defined at every time instant during the transformation).
* A transformation is said to be **physically possible** if it does not violate the conservation of energy and the second law of thermodynamics expressed by the Kelvin-Planck statement, or equivalently, by the Clausius statement.
* A transformation F is said to be reversible if there exists a **physically possible** transformation G such that 1) the initial state of G equals the final state of F; 2) the final state of G equals the initial state of F; 3) the heat $Q\_F$ exchanged by the system during F equals the opposite of the heat $Q\_G$ exchanged during G (that is $Q\_F = -Q\_C$); and 4) the work $W\_F$ exchanged by the system during F equals the opposite of the work $W\_G$ exchanged during G.

My question is: is my definition of reversible transformation well-posed? If it is, then how can I **logically and rigorously** prove that if a transformation is reversible then it has to be quasistatic **without** using the concept of entropy, but using only the Kelvin-Planck and Clausius versions of the second law of thermodynamics?

I know how to prove that not all quasistatic transformations are reversible with the previous definitions, so I am asking for a **logical and rigorous** proof for the opposite: all reversible transformations are quasistatic.

If my definition of reversibility is wrong, then what is a definition of reversible transformations that does not refer to "quasistaticity" and does not use the concept of entropy such that I can prove that all reversible transformations are quasistatic, again, without using entropy?