reserve x,y,X,Y for set,
  k,l,n for Nat,
  i,i1,i2,i3,j for Integer,
  G for Group,
  a,b,c,d for Element of G,
  A,B,C for Subset of G,
  H,H1,H2, H3 for Subgroup of G,
  h for Element of H,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem
  G is commutative Group implies for P being Permutation of dom F1 st F2
  = F1 * P holds Product(F1) = Product(F2)
proof
  set g = the multF of G;
  assume G is commutative Group;
  then g is commutative by GROUP_3:2;
  hence thesis by FINSOP_1:7;
end;
