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 Th9:
  (F1 |^ a) ^ (F2 |^ a) = (F1 ^ F2) |^ a
proof
A1: now
    let k;
    assume k in dom(F1 |^ a);
    then k in Seg len(F1 |^ a) by FINSEQ_1:def 3;
    then k in Seg len F1 by Def1;
    then
A2: k in dom F1 by FINSEQ_1:def 3;
    then
A3: F1/.k = (F1 ^ F2)/.k & k in dom(F1 ^ F2) by FINSEQ_3:22,FINSEQ_4:68;
    thus (F1 |^ a).k = (F1/.k) |^ a by A2,Def1
      .= ((F1 ^ F2) |^ a).k by A3,Def1;
  end;
A4: now
    let k;
    assume
A5: k in dom(F2 |^ a);
    len F2 = len(F2 |^ a) by Def1;
    then
A6: k in dom F2 by A5,FINSEQ_3:29;
    then len F1 + k in dom(F1 ^ F2) by FINSEQ_1:28;
    then len(F1 |^ a) + k in dom(F1 ^ F2) by Def1;
    hence
    ((F1 ^ F2) |^ a).(len(F1 |^ a) + k) = ((F1 ^ F2)/.(len(F1 |^ a) + k))
    |^ a by Def1
      .= ((F1 ^ F2)/.(len F1 + k)) |^ a by Def1
      .= (F2/.k) |^ a by A6,FINSEQ_4:69
      .= (F2 |^ a).k by A6,Def1;
  end;
  len(F1 |^ a) + len(F2 |^ a) = len F1 + len(F2 |^ a) by Def1
    .= len F1 + len F2 by Def1
    .= len(F1 ^ F2) by FINSEQ_1:22
    .= len((F1 ^ F2) |^ a) by Def1;
  hence thesis by A1,A4,FINSEQ_3:38;
end;
