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