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 Th27:
  [.a",b.] = [.b,a.] |^ a"
proof
  thus [.a",b.] = a"" * (b" * (a" * b)) by Th16
    .= a"" * (b" * (a" * b)) * 1_G by GROUP_1:def 4
    .= a"" * (b" * (a" * b)) * (a * a") by GROUP_1:def 5
    .= a"" * (b" * (a" * b)) * a * a" by GROUP_1:def 3
    .= a"" * ((b" * (a" * b)) * a) * a" by GROUP_1:def 3
    .= [.b,a.] |^ a" by Th16;
end;
