 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 Th21:
  for a, b holds Conjugate (a * b) = (Conjugate b) * (Conjugate a)
proof
  let a, b;
  set f1 = Conjugate (a * b);
  set f2 = ((Conjugate b) * (Conjugate a));
A1: for c holds f1.c = c |^ a |^ b
  proof
    let c;
    c |^ (a * b) = c |^ a |^ b by GROUP_3:24;
    hence thesis by Def6;
  end;
A2: for c holds f1.c = (Conjugate b).(c |^ a)
  proof
    let c;
    c |^ a |^ b = (Conjugate b).(c |^ a) by Def6;
    hence thesis by A1;
  end;
A3: for c holds f1.c = (Conjugate b).((Conjugate a).c)
  proof
    let c;
    (Conjugate b).(c |^ a) = (Conjugate b).((Conjugate a).c) by Def6;
    hence thesis by A2;
  end;
  for c holds f1.c = f2.c
  proof
    let c;
    (Conjugate b).((Conjugate a).c) = f2.c by FUNCT_2:15;
    hence thesis by A3;
  end;
  hence thesis;
end;
