 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
  for a holds Conjugate a" = (Conjugate a)"
proof
  let a;
  set f = Conjugate a;
  set g = Conjugate a";
A1: g * f = Conjugate 1_G by Th25
    .= id the carrier of G by Th22;
A2: f is Element of Aut G by Th12;
  then f is Homomorphism of G, G by Def1;
  then
A3: f is one-to-one by A2,Def1;
  rng f = dom f by A2,Lm3
    .= the carrier of G by A2,Lm3;
  hence thesis by A1,A3,FUNCT_2:30;
end;
