
theorem
  {i where i is Nat : ArProg (1,2).i is triangular} is infinite
  proof
    set X = {i where i is Nat : ArProg (1,2).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 + 1) is triangular by NUMPOLY1:def 2;
      Triangle (4 * m + 1) = (4 * m + 1) * (4 * m+1+1) / 2 by NUMPOLY1:19
                    .= 2 * (m*(4*m) + 3*m) + 1; then
A4:   Triangle (4*m+1) = ArProg (1,2).(m*(4*m)+3*m) by ArDefNth;
      set n = m * (4 * m) + 3 * m;
      4 * m + 3 >= 1 by NAT_1:12; then
a1:   m * (4 * m + 3) >= m * 1 by NAT_1:4;
      n in X by A3,A4;
      hence thesis by a1;
    end;
    hence thesis by PYTHTRIP:9;
  end;
