Now, the question was indeed to show that language C is not in RE. We do that by a reduction from the COMPLEMENT of A_TM.

The reduction is the following: f(<M,w>)=M'. M' on input z: if z equals w then simulate M(w) and accept if M halts. Otherwise accept.

It is easy to see that if M does not halt on W then there are two different words for which M' behaves differently and if M halts on w then M' behaves the same on all inputs.

Good Luck!

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