[SOLVED] Understanding a proof of Carnot's theorem.

I'm having trouble understand this proof of Carnot's theorem, and I would appreciate it if someone could point out where my reasoning goes wrong.

The proof reads thusly:

My problem with it is that is seems to imply that either the second law is wrong, or the Carnot engine is not the most efficient because:

Why not just get an engine that is less efficient than the Carnot one and construct the same device as was made in the proof except with the Carnot engine in the place that the hypothetical one occupied in the proof, and the less efficient engine in the place of the Carnot?

This, as far as I can see, would also move heat from the cold reservoir to the hot one.

that's not efficiency,that's co-efficient of performance..
we are working with efficiency here so if we take coefficient of performance we must have to convert it into efficiency through the relation between them finally..:)