Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work (the drained heat is fully converted to work) in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the right figure. The efficiency of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/ η. The net and sole effect of the combined pair of engines is to transfer heat Δ Δ Q = Q ( 1 η η − − 1 ) from the cooler reservoir to the hotter one, which violates the Clausius statement. This is a consequence of the first law of thermodynamics, as for the total system's energy to remain the same; Input + Output = 0 ⟹ ⟹ ( Q + Q c ) − − Q η η = 0 , so therefore Q c = Q ( 1 η η − − 1 ) , where (1) the sign convention of heat is used in which heat entering into (leaving from) an engine is positive (negative) and (2) Q η η is obtained by the definition of efficiency of the engine when the engine operation is not reversed. Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.