let G be Group; :: thesis: for F1, F2 being FinSequence of the carrier of G st G is commutative Group holds
for P being Permutation of (dom F1) st F2 = F1 * P holds
Product F1 = Product F2
let F1, F2 be FinSequence of the carrier of G; :: thesis: ( G is commutative Group implies for P being Permutation of (dom F1) st F2 = F1 * P holds
Product F1 = Product F2 )
set g = the multF of G;
A1: the multF of G is associative by GROUP_1:31;
assume G is commutative Group ; :: thesis: for P being Permutation of (dom F1) st F2 = F1 * P holds
Product F1 = Product F2
then A2: the multF of G is commutative by GROUP_3:2;
let P be Permutation of (dom F1); :: thesis: ( F2 = F1 * P implies Product F1 = Product F2 )
assume F2 = F1 * P ; :: thesis: Product F1 = Product F2
hence Product F1 = Product F2 by A2, A1, FINSOP_1:8, GROUP_1:34; :: thesis: verum