 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 Th10:
  for f being Element of Aut G for g being Element of AutGroup G
  st f = g holds f" = g"
proof
  let f be Element of Aut G;
  let g be Element of AutGroup G;
  consider g1 be Element of Aut G such that
A1: g1 = g";
  assume f = g;
  then g1 * f = g" * g by A1,Def2;
  then g1 * f = 1_AutGroup G by GROUP_1:def 5;
  then
A2: g1 * f = id the carrier of G by Th9;
  f is Homomorphism of G, G by Def1;
  then
A3: f is one-to-one by Def1;
  rng f = dom f by Lm3
    .= the carrier of G by Lm3;
  hence thesis by A1,A3,A2,FUNCT_2:30;
end;
