Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Open Scope nat_scope.
Definition sum_of_two_square :=
[qualify a x |
[exists a : 'I_x.+1, exists b : 'I_x.+1, x == a ^ 2 + b ^ 2]].
Fact sum2s0 : 0 \is a sum_of_two_square.

Lemma sum2sX_prime x n :
prime x -> odd n ->
x ^ n \is  a sum_of_two_square -> x  \is  a sum_of_two_square.