Proof for "It is my contention that these operations include all those which are used in the computation of a number"
A contention is something that one asserts to be true. Turing's contention here is essentially the Church&ndash;Turing thesis: that Turing machines can compute everything that can be reasonably computed. This isn't something that can be proven without a definition of "reasonably computed". But, even if one were to define that term, it would be arguable that the definition wasn't a good one and that, again, isn't something that can be proven.

Turing's paper includes a lot of material about computable sequences of numbers and also a sketch proof that Turing machines and the lambda calculus are equally powerful. That constitutes quite a lot of evidence for the original contention, though I didn't notice Turing explicitly mentioning this when I skimmed the paper just now.

In particular, one might choose to define "reasonably computable" as meaning "computable by any possible physical system." Well, Turing machines can simulate every physical process we're aware of, including quantum mechanics. But, of course, we can't prove that there are no physical processes we don't know about yet that can't be simulated by Turing machines.