theorem
  a |^ b = a * [.a,b.]
proof
  a * [.a,b.] = a * ((a" * b") * (a * b)) by GROUP_1:def 3
             .= a * (a" * (b" * (a * b))) by GROUP_1:def 3
             .= (a * a") * (b" * (a * b)) by GROUP_1:def 3
             .= 1_G * (b" * (a * b)) by GROUP_1:def 5
             .= b" * (a * b) by GROUP_1:def 4
             .= b" * a * b by GROUP_1:def 3;
  hence thesis;
end;
