reserve X for set;
reserve G for Group;
reserve H for Subgroup of G;
reserve h,x,y for object;
reserve f for Endomorphism of G;
reserve phi for Automorphism of G;

theorem
  G is finite implies
  for p being prime Nat
  for P being strict Subgroup of G
  st P is_Sylow_p-subgroup_of_prime p
  holds Image(phi|P) is_Sylow_p-subgroup_of_prime p
proof
  assume A0: G is finite;
  let p be prime Nat;
  let P be strict Subgroup of G;
  assume A1: P is_Sylow_p-subgroup_of_prime p;
  then A2: P is p-group by GROUP_10:def 18;
  set Q = phi .: P;
  consider r being Nat such that
  A3: card P = p |^ r by A2,GROUP_10:def 17;
  card Q = p |^ r by A3,Th19,GROUP_6:75;
  then A4: Q is p-group by GROUP_10:def 17;
  A5: Q = Image(phi|P) by GRSOLV_1:def 3;
  not p divides index P by A1, GROUP_10:def 18;
  then not p divides index Q by A0,Th19,Th20;
  hence Image(phi|P) is_Sylow_p-subgroup_of_prime p by A4, A5, GROUP_10:def 18;
end;
