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

theorem Th3:
  permutations(Seg n) = Permutations(n)
  proof
    thus permutations(Seg n) c= Permutations(n)
    proof
      let x be object;
      assume x in permutations(Seg n);
      then x is Permutation of Seg n by Th1;
      hence thesis by MATRIX_1:def 12;
    end;
    let x be object;
    assume x in Permutations(n);
    then x is Permutation of Seg n by MATRIX_1:def 12;
    hence thesis;
  end;
