 reserve n,s for Nat;

theorem Th24:
  for n being triangular number holds
    not n, 2 are_congruent_mod 10
  proof
    let n be triangular number;
    consider k being Nat such that
A1: n = Triangle k by Def2;
A2: 4 * 5 = 20;
A3: n = k * (k + 1) / 2 by A1,Th19;
    assume n, 2 are_congruent_mod 10; then
    k * (k + 1) / 2 * 2, 2 * 2 are_congruent_mod (10 * 2)
      by A3,INT_4:10; then
    k * (k + 1), 4 are_congruent_mod 5 by A2,INT_1:20; then
    4, k * k + k are_congruent_mod 5 by INT_1:14;
    hence thesis by Th6,INT_1:14;
  end;
