reserve G for Group;
reserve A,B for non empty Subset of G;
reserve N,H,H1,H2 for Subgroup of G;
reserve x,a,b for Element of G;

theorem Th5:
  for N be Subgroup of G
   st for x,y being Element of G st y in N holds x * y * x" in N
   holds N is normal
proof
  let N be Subgroup of G such that
A1:for x,y being Element of G st y in N holds x * y * x" in N;
  for x be Element of G holds x * N c= N * x
  proof
    let x be Element of G;
    let z be object;
    assume
A2:z in x * N;
    then reconsider z as Element of G;
    consider z1 be Element of G such that
A3: z = x * z1 & z1 in N by A2,GROUP_2:103;
A4: x * z1 * x" in N by A1,A3;
    (x * z1 * x") * x = z by A3,GROUP_3:1;
    hence thesis by A4,GROUP_2:104;
  end;
  hence thesis by GROUP_3:118;
end;
