reserve X,Y for set;
reserve G for Group;
reserve n for Nat;

theorem Th6:
  1_SymGroup(X) = id X
  proof
    set e = id X;
    e in permutations(X);
    then reconsider e as Element of SymGroup(X) by Def2;
    now
      let h be Element of SymGroup(X);
      reconsider h1 = h as Permutation of X by Th5;
      thus h * e = e * h1 by Def2
      .= h by FUNCT_2:17;
      thus e * h = h1 * e by Def2
      .= h by FUNCT_2:17;
    end;
    hence thesis by GROUP_1:4;
  end;
