
theorem
  {i where i is Nat : ArProg (0,2).i is triangular} is infinite
  proof
    set X = {i where i is Nat : ArProg (0,2).i is triangular};
    for m being Nat ex n be Nat st n >= m & n in X
    proof
      let m be Nat;
A3:   Triangle (4 * m) is triangular by NUMPOLY1:def 2;
A2:   Triangle (4*m) = (4*m) * (4*m+1) / 2 by NUMPOLY1:19
                    .= (2*m) * (4*m+1);
a4:   ArProg (0,2).(m*(4*m+1)) = 0 + 2 * (m * (4 * m + 1)) by ArDefNth;
      4 * m + 1 >= 1 by NAT_1:12; then
AA:   m * (4 * m + 1) >= m * 1 by NAT_1:4;
      m * (4 * m + 1) in X by A3,a4,A2;
      hence thesis by AA;
    end;
    hence thesis by PYTHTRIP:9;
  end;
