How do you define "prime cardinal" without relying on the definition of prime numbers? "Prime" (as distinct from irreducible elements) doesn't have a clear meaning before you prove the Fundamental Theorem of Arithmetic, does it?
I defined it in the relevant way in the post you're responding to (indeed, inbetween the very words "prime" and "cardinal" where I first invoke the concept): the relevant notion is of a cardinal which can't be factored as the (Cartesian) product of cardinals other than 1 and itself. [Yes, people often call this concept "irreducible" instead, but I used "prime" for convenience, and explicitly described what I meant by that.]
This is the concept that is relevant to the provided proof: Gowers argues that, if a group's size is an unfactorable cardinal, then, by Lagrange's Theorem (which tells us |G'| divides |G| whenever G' is a subgroup of G), it has no intermediate subgroups. Thus, if the additive group mod whatever has size an unfactorable cardinal, then every homomorphism into it is either constantly identity or surjective (as its range is a subgroup); accordingly, multiplication by any non-identity element would have to be invertible (modulo whatever), which is the desired instance of Bezout's Theorem.
