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