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 Th15:
  (F |^ a) |^ I = (F |^ I) |^ a
proof
  len(F |^ I) = len F by GROUP_4:def 3;
  then
A1: dom(F |^ I) = dom F by FINSEQ_3:29;
A2: len(F |^ a) = len F by Def1;
  then
A3: dom(F |^ a) = dom F by FINSEQ_3:29;
A4: len((F |^ a) |^ I) = len(F |^ a) by GROUP_4:def 3;
  then
A5: dom ((F |^ a) |^ I) = Seg len F by A2,FINSEQ_1:def 3;
A6: now
    let k be Nat;
    assume k in dom ((F |^ a) |^ I);
    then
A7: k in dom F by A5,FINSEQ_1:def 3;
    then
A8: (F |^ a)/.k = (F |^ a).k by A3,PARTFUN1:def 6;
A9: (F |^ I)/.k = (F |^ I).k by A1,A7,PARTFUN1:def 6;
    thus ((F |^ a) |^ I).k = ((F |^ a)/.k) |^ @(I/.k) by A3,A7,GROUP_4:def 3
      .= ((F/.k) |^ a) |^ @(I/.k) by A7,A8,Def1
      .= ((F/.k) |^ @(I/.k)) |^ a by GROUP_3:28
      .= ((F |^ I)/.k) |^ a by A7,A9,GROUP_4:def 3
      .= ((F |^ I) |^ a).k by A1,A7,Def1;
  end;
  len(F |^ I |^ a) = len(F |^ I) by Def1
    .= len F by GROUP_4:def 3;
  hence thesis by A2,A4,A6,FINSEQ_2:9;
end;
