Answer
See answer below.
Work Step by Step
The Kelvin–Planck definition says that it's impossible for there to be adevice that converts 100% of the input input energy into work, since some heat will always be lost to the environment (friction, faulty insulation, etc.)
The Clausius definition says that's not possible to create a device that only transfer heat from a colder source to a warmer one, requiring input energy to work.
For a cycle that violates the first definition $W_e=Q_H$, and uses this work in a refrigerator, rejecting $Q_H+Q_L$ to the hot reservoir, would be a violation of the second definition too, hence they are equivalent.
