theorem Th9:
  id the carrier of G = 1_AutGroup G
proof
  set f = the Element of AutGroup G;
  reconsider g = id the carrier of G as Element of AutGroup G by Th3;
  consider g1 be Function of the carrier of G, the carrier of G such that
A1: g1 = g;
  f is Homomorphism of G, G by Def1;
  then consider f1 be Function of the carrier of G, the carrier of G such that
A2: f1 = f;
  f1 = f & g1 = g implies f1 * g1 = f * g by Def2;
  hence thesis by A1,A2,FUNCT_2:17,GROUP_1:7;
end;
