reserve x,y for set,
  G for Group,
  A,B,H,H1,H2 for Subgroup of G,
  a,b,c for Element of G,
  F,F1 for FinSequence of the carrier of G,
  I,I1 for FinSequence of INT,
  i,j for Element of NAT;

theorem Th9:
  [.A,B.] is Subgroup of [.B,A.]
proof
  now let a;
    assume a in [.A,B.];
    then consider F,I such that len F = len I and
A1: rng F c= commutators(A,B) and
A2: a = Product(F |^ I) by GROUP_5:61;
    deffunc F(Nat) = (F/.$1)";
    consider F1 such that
A3: len F1 = len F and
A4: for k be Nat st k in dom F1 holds F1.k = F(k) from FINSEQ_2:sch 1;
A5: dom F1 = Seg len F by A3,FINSEQ_1:def 3;
    deffunc F(Nat) = @(- @(I/.$1));
    consider I1 such that
A6: len I1 = len F and
A7: for k be Nat st k in dom I1 holds I1.k = F(k) from FINSEQ_2:sch 1;
A8: dom F = Seg len F by FINSEQ_1:def 3;
A9: dom F1 = dom F by A3,FINSEQ_3:29;
A10: dom I1 = dom F by A6,FINSEQ_3:29;
    set J = F1 |^ I1;
A11: len J = len F & len(F |^ I) = len F by A3,GROUP_4:def 3;
   then
A12: dom(F |^ I)  = Seg len F by FINSEQ_1:def 3;
    now
      let k be Nat;
      assume
A13:  k in dom(F |^ I); then
A14:  k in dom F by A12,FINSEQ_1:def 3;
      J.k = (F1/.k) |^ @(I1/.k) & F1/.k = F1.k & F1.k = (F/.k)" &
      I1.k = (I1/.k) & @(I1/.k) = I1/.k &
      I1.k = @(- @(I/.k)) & @(- @(I/.k)) = - @(I/.k)
        by A4,A7,A8,A9,A10,A13,A12,GROUP_4:def 3,PARTFUN1:def 6;
      then J.k = ((F/.k)" |^ @(I/.k))" by GROUP_1:36
        .= ((F/.k) |^ @(I/.k))"" by GROUP_1:37
        .= (F/.k) |^ @(I/.k);
      hence (F |^ I).k = J.k by A14,GROUP_4:def 3;
    end; then
A15: J = F |^ I by A11,FINSEQ_2:9;
    rng F1 c= commutators(B,A)
    proof
      let x be object;
      assume x in rng F1;
      then consider y being object such that
A16:  y in dom F1 and
A17:  F1.y = x by FUNCT_1:def 3;
      reconsider y as Element of NAT by A16;
      y in dom F by A3,A16,FINSEQ_3:29;
      then F.y in rng F by FUNCT_1:def 3;
      then consider b,c such that
A18:  F.y = [.b,c.] and
A19:  b in A & c in B by A1,GROUP_5:52;
      x = (F/.y)" & F/.y = F.y by A16,A4,A8,A17,A5,PARTFUN1:def 6;
      then x = [.c,b.] by A18,GROUP_5:22;
      hence thesis by A19,GROUP_5:52;
    end;
    hence a in [.B,A.] by A2,A3,A6,A15,GROUP_5:61;
  end;
  hence thesis by GROUP_2:58;
end;
