reserve x,y for set,
  k,n for Nat,
  i for Integer,
  G for Group,
  a,b,c ,d,e for Element of G,
  A,B,C,D for Subset of G,
  H,H1,H2,H3,H4 for Subgroup of G ,
  N1,N2 for normal Subgroup of G,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem Th14:
  Product(F |^ a) = Product(F) |^ a
proof
  defpred P[FinSequence of the carrier of G] means Product($1 |^ a) = Product(
  $1) |^ a;
A1: now
    let F,b;
    assume
A2: P[F];
    Product((F ^ <* b *>) |^ a) = Product((F |^ a) ^ (<* b *> |^ a)) by Th9
      .= Product(F) |^ a * Product(<* b *> |^ a) by A2,GROUP_4:5
      .= Product(F) |^ a * Product<* b |^ a *> by Th11
      .= Product(F) |^ a * (b |^ a) by FINSOP_1:11
      .= (Product(F) * b) |^ a by GROUP_3:23
      .= Product(F ^ <* b *>) |^ a by GROUP_4:6;
    hence P[F^<*b*>];
  end;
A3: P[<*> the carrier of G]
  proof
    set p = <*> the carrier of G;
    thus Product(p |^ a) = Product p by Th10
      .= 1_G by GROUP_4:8
      .= (1_G) |^ a by GROUP_3:17
      .= Product p |^ a by GROUP_4:8;
  end;
  for F holds P[F] from FINSEQ_2:sch 2(A3,A1);
  hence thesis;
end;
