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" |^ 2) * ((a * b") |^ 2)* (b |^ 2)
proof
  thus [.a,b.] = (a" * b") * (a * b) by Th16
    .= (a" * 1_G * b") * (a * b) by GROUP_1:def 4
    .= (a" * 1_G * b") * (a * 1_G * b) by GROUP_1:def 4
    .= (a" * (a" * a) * b") * (a * 1_G * b) by GROUP_1:def 5
    .= (a" * (a" * a) * b") * (a * (b" * b) * b) by GROUP_1:def 5
    .= (a" * a" * a * b") * (a * (b" * b) * b) by GROUP_1:def 3
    .= (a" |^ 2 * a * b") * (a * (b" * b) * b) by GROUP_1:27
    .= (a" |^ 2 * a * b") * (a * ((b" * b) * b)) by GROUP_1:def 3
    .= (a" |^ 2 * a * b") * (a * (b" * (b * b))) by GROUP_1:def 3
    .= (a" |^ 2 * a * b") * (a * (b" * (b |^ 2))) by GROUP_1:27
    .= (a" |^ 2 * (a * b")) * (a * (b" * (b |^ 2))) by GROUP_1:def 3
    .= (a" |^ 2) * ((a * b") * (a * (b" * (b |^ 2)))) by GROUP_1:def 3
    .= (a" |^ 2) * ((a * b") * (a * b" * (b |^ 2))) by GROUP_1:def 3
    .= (a" |^ 2) * ((a * b") * (a * b") * (b |^ 2)) by GROUP_1:def 3
    .= (a" |^ 2) * ((a * b") * (a * b")) * (b |^ 2) by GROUP_1:def 3
    .= (a" |^ 2) * ((a * b") |^ 2) * (b |^ 2) by GROUP_1:27;
end;
