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;
reserve K for characteristic Subgroup of G;

theorem Th35:
  K is normal Subgroup of G
proof
  for a being Element of G holds K |^ a = the multMagma of K
  proof
    let a be Element of G;
    consider f being inner Automorphism of G such that
    A2: a is_inner_wrt f by Th32;
    the multMagma of K = Image(f|K) by Def3
                      .= K |^ a by A2,Th28;
    hence thesis;
  end;
  hence K is normal Subgroup of G by GROUP_3:def 13;
end;
