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 Th22:
  [.a,b.]" = [.b,a.]
proof
  thus [.a,b.]" = ((a" * b") * (a * b))" by Th16
    .= (a * b)" * (a" * b")" by GROUP_1:17
    .= (b" * a") * (a" * b")" by GROUP_1:17
    .= (b" * a") * (b"" * a"") by GROUP_1:17
    .= (b" * a") * (b"" * a)
    .= (b" * a") * (b * a)
    .= [.b,a.] by Th16;
end;
