:: deftheorem Def44 defines normalAIMLOOP:def 46 : for Q being multLoop for H being SubLoop of Q holds ( H is normal iff for x, y being Element of Q holds ( (x * H)*(y * H)=(x * y)* H & ( for z being Element of Q holds ( ( (x * H)*(y * H)=(x * H)*(z * H) implies y * H = z * H ) & ( (y * H)*(x * H)=(z * H)*(x * H) implies y * H = z * H ) ) ) ) );