 reserve G for Group;
 reserve H for Subgroup of G;
 reserve a, b, c, x, y for Element of G;
 reserve h for Homomorphism of G, G;
 reserve q, q1 for set;

theorem
  G is commutative Group implies
  for f being Element of InnAutGroup G holds f = 1_InnAutGroup G
proof
  assume
A1: G is commutative Group;
  let f be Element of InnAutGroup G;
  the carrier of InnAutGroup G = InnAut G by Def5;
  then consider f1 be Element of InnAut G such that
A2: f1 = f;
  f1 is Element of Funcs (the carrier of G, the carrier of G) by FUNCT_2:8;
  then consider a such that
A3: for x holds f1.x = x |^ a by Def4;
A4: for x holds f1.x = x
  proof
    let x;
    thus f1.x = x |^ a by A3
      .= x by A1,GROUP_3:29;
  end;
  for x holds f1.x = (id the carrier of G).x by A4;
  then f1 = id the carrier of G;
  hence thesis by A2,Th19;
end;
