As with a lot of proofs by contradiction the fist line "assume that something is not..." is not necessary.
f injective iff ker(f)={e} ( the => is easy since it is a special case of the the general property of being injective: f(e)=e so if anything else satisfies f(g)=e thetn g=e and ker(f)={e})
f(x)=f(y) implies f(xy^{1}=e, which, by hypothesis means xy^{1}=e as we required to show.
so we don't prove it by assuming the opposite and getting a contradiction since the proof is actually direct. the same can be done for cantor's diagonal argument.
