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
  [.a,b,b |^ a.] = [.b,[.b,a.].]
proof
  thus [.a,b,b |^ a.] = [.b,a.] * (b |^ a)" * [.a,b.] * (b |^ a) by Th22
    .= (b" * a" * b * a) * (b" |^ a) * [.a,b.] * (b |^ a) by GROUP_3:26
    .= (b" * a" * b * a) * (a" * (b" * a)) * [.a,b.] * (b |^ a) by
GROUP_1:def 3
    .= (b" * a" * b * a) * a" * (b" * a) * [.a,b.] * (b |^ a) by GROUP_1:def 3
    .= (b" * a" * b) * (b" * a) * [.a,b.] * (b |^ a) by GROUP_3:1
    .= b" * a" * (b * (b" * a)) * [.a,b.] * (b |^ a) by GROUP_1:def 3
    .= b" * a" * a * [.a,b.] * (b |^ a) by GROUP_3:1
    .= b" * (a"* b"* a * b) * (b |^ a) by GROUP_3:1
    .= b" * (a"* b"* a * b) * 1_G * (a" * b * a) by GROUP_1:def 4
    .= b" * [.a,b.] * (b * b") * (a" * b * a) by GROUP_1:def 5
    .= b" * [.a,b.] * ((b * b") * (a" * b * a)) by GROUP_1:def 3
    .= b" * [.a,b.] * (b * (b" * (a" * b * a))) by GROUP_1:def 3
    .= b" * [.a,b.] * (b * [.b,a.]) by Th16
    .= b" * [.b,a.]" * (b * [.b,a.]) by Th22
    .= [.b,[.b,a.].] by Th16;
end;
