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;
