 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);
 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);
reserve G1,G2 for Group;

theorem Th69:
  Aut (1).G = {id (1).G}
proof
  for x being object st x in {id (1).G} holds x in Aut (1).G
  proof
    let x be object;
    assume x in {id (1).G};
    then x = id the carrier of (1).G by TARSKI:def 1;
    then x is Element of Aut (1).G by AUTGROUP:3;
    hence x in Aut (1).G;
  end;
  then A1: {id (1).G} c= Aut (1).G;
  for x being object st x in Aut (1).G holds x in {id (1).G}
  proof
    let x be object;
    assume x in Aut (1).G;
    then reconsider phi=x as Homomorphism of (1).G,(1).G by AUTGROUP:def 1;
    for g being Element of (1).G
    holds (id (1).G).g = phi.g
    proof
      let g be Element of (1).G;
      g in (1).G;
      then g in {1_G} by GROUP_2:def 7;
      then A2: g = 1_G by TARSKI:def 1;
      hence phi.g = phi.(1_((1).G)) by GROUP_2:44
                 .= 1_((1).G) by GROUP_6:31
                 .= 1_G by GROUP_2:44
                 .= (id (1).G).g by A2;
    end;
    then id (1).G = x by FUNCT_2:def 8;
    hence x in {id (1).G} by TARSKI:def 1;
  end;

  then Aut (1).G c= {id (1).G};
  hence Aut (1).G = {id (1).G} by A1, XBOOLE_0:def 10;
end;
