Identification of parameters of Poisson distributions by the extreme order statistics

Let Poisson variables X-1, ..., X-m be independent and Y-1, ..., Y-n be also independent such that each X-i similar to P(lambda(i)), Y-J similar to P(mu(j)). We prove that if X-m:m = max{X-1, ..., X-m} and Y-n:n = max{Y-1, ..., Y-n} have the same distribution, then m = n and {lambda(1), ..., lambda(m)} = {mu(1), ..., mu(m)}: The weak version in term of minimum statistics is also derived.