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