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 |^ n,b.] = a |^ (- n) * ((a |^ b) |^ n)
proof
  thus [.a |^ n,b.] = (a |^ n)" * (b" * (a |^ n) * b) by Th16
    .= a |^ (- n) * ((a |^ n) |^ b) by GROUP_1:36
    .= a |^ (- n) * ((a |^ b) |^ n) by GROUP_3:27;
end;
