Equivalent of Kelvin Plank and Clausius statement:

The Kelvin plank and clausius statements of the second law and are equivalent in all the respect. The equivalence of statement will be proved by logic and violation of one statement leads to violation of second statement and vice versa of it.

Violation of Clausius statement:

A cyclic HP transfer heat from the cold reservoir(T₂) to the hot reservoir(T₁) having no work input. This violates the clausius statement.

A Cyclic HE operates in between the same reservoirs drawing heat Q₁ and producing W as work. As HP is supplying Q₁ heat to the hot reservoir, the hot reservoir can be removed. The HP and HE comprise of HE operating in cycle and generating work W while exchanging heat with one reservoir(Cold) only. This violates K-P statement