Bill Theorem 1, There can be no engine that produces less work from heat,
than produced by a reversible engine, whether the engine is reversible or irreversible.
and Corollaries

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Complementary of Carnot/Kelvin's Theorem of Maximum Work from a Heat Engine.

Summarized

The Bill Theorem of Minimum Work output of Heat Engines

Statement: No heat engines of any type may produce less than the amount of work returned for a unit of heat, than the work returned by a reversible heat engine.

Proof: Should any heat engine produce less than the work output for a given unit of heat, produced by a reversible engine, the reversible engine could be used to convert the entirety of the lesser work output back to heat. This would return the heat energy to less than its original state, and leave no work energy or any other energy produced

This amount to destruction of, disappearance of, or conversion of energy to nothing, a violation of the Law of Conservation of Energy.

The Law of Conservation of Energy requires equal energy conversion of all types.

More specifically, the current theory of Heat is molecular motion. Transfer from Heat to and from Machine, is a transfer of momentum. This proof relies only on the more narrow law of Conservation of Momentum.

Bill Corollary, Irreversible Heat Engine efficiency

Bill Corollary 1, Minimum Heat Output of Irreversible Engines

Statement: No irreversible heat engines of any type may produce less than the amount of work returned for a unit of heat, than the work returned by a reversible heat engine.

Proof: Although no heat engines other than Vapor Expansion engines are identified, postulate there is an irreversible heat engine yet undiscovered.

Although reversibility would prove the Law of Conservation of Energy is obeyed. The lack of reversibility does not prove the Conservation of Energy is disobeyed.

By the same argument as the Theorem of Minimum output of Heat Engines, irreversible Heat Engines must produce the same quantity of Work for a given quantity of Heat as a Reversible Heat Engine.

Bill Corollary 2, Equal Work Output of all Heat Engines

Statement: All Heat Engines of any type produce the same amount of Work for a given amount of Heat.

This is a corollary to both the Carnot/Kelvin Theorem of Maximum Work output of a heat engine, and the Bill Theorem of Minimum Work output of a Heat Engine.

Proof: Conversion of Work To or From Heat must obey the Law of Conservation of Energy, which for Thermodynamic engines is simply stated as Energy = Heat + Work.

Bill Corollary 3, Heat is an illusion. Heat engines are momentum transfer devices between molecular momentum of gas, and machine momentum.

The proof of this is the current theory of Heat as Motion, since about 1840. The energy in heat is the energy of motion, not actually a different form of energy. Consequently, even if the general law of Conservation of Energy is ever proved invalid, the conclusions of the above theorems and corollaries only relies on Conservation of Momentum, and Newton's laws of motion.

It is notable that that Newton's laws of motion were established by the time of Sadi Carnot's work. Had the theory of Heat as molecular motion been the dominant theory during his work (instead of Heat as an imponderable material), it seems probable he would have made the same connection.