reserve L for non empty addLoopStr;
reserve a,b,c,x for Element of L;
reserve L for non empty multLoopStr;
reserve a,b,c,x,y,z for Element of L;

theorem
  L is commutative multGroup iff (for a holds a * 1.L = a) & (for a ex x
  st a*x = 1.L) & (for a,b,c holds (a*b)*c = a*(b*c)) & for a,b holds a*b = b*a
  by Th11,GROUP_1:def 12;
