Dear tutor, this is advanced math, may you answer this question (a,b,c,d,e) for me with steps and explain the work please. thank you

33. (a) Prove that n + m is defined for all n, me P. (b) Prove that n + 1 = 1 + n for all ne P. (c) Prove that m' + n = (m + n)' for all m, ne P. (d) Prove that addition is commutative. That is, prove that n + m = m + n for all m, n E P. (e) Prove that addition is associative. That is, prove that (m + n) + p m + (n + p) for all m, n, pe P