theorem
  [.a,a".] = 1_G & [.a",a.] = 1_G
proof
  thus [.a,a".] = (a" * a"") * (a * a") by Th16
    .= 1_G * (a * a") by GROUP_1:def 5
    .= a * a" by GROUP_1:def 4
    .= 1_G by GROUP_1:def 5;
  thus [.a",a.] = (a"" * a") * (a" * a) by Th16
    .= (a"" * a") * 1_G by GROUP_1:def 5
    .= a"" * a" by GROUP_1:def 4
    .= 1_G by GROUP_1:def 5;
end;
