 reserve n,s for Nat;

theorem
  for n being triangular number holds
    n, 0 are_congruent_mod 10 or
    n, 1 are_congruent_mod 10 or
    n, 3 are_congruent_mod 10 or
    n, 5 are_congruent_mod 10 or
    n, 6 are_congruent_mod 10 or
    n, 8 are_congruent_mod 10
  proof
    let n be triangular number;
    n, 0 are_congruent_mod 10 or ... or n, 9 are_congruent_mod 10 by Th9;
    then per cases;
    suppose
      n, 0 are_congruent_mod 10;
      hence thesis;
    end;
    suppose
      n, 1 are_congruent_mod 10;
      hence thesis;
    end;
    suppose
      n, 2 are_congruent_mod 10;
      hence thesis by Th24;
    end;
    suppose
      n, 3 are_congruent_mod 10;
      hence thesis;
    end;
    suppose
      n, 4 are_congruent_mod 10;
      hence thesis by Th25;
    end;
    suppose
      n, 5 are_congruent_mod 10;
      hence thesis;
    end;
    suppose
      n, 6 are_congruent_mod 10;
      hence thesis;
    end;
    suppose
      n, 7 are_congruent_mod 10;
      hence thesis by Th22;
    end;
    suppose
      n, 8 are_congruent_mod 10;
      hence thesis;
    end;
    suppose
      n, 9 are_congruent_mod 10;
      hence thesis by Th23;
    end;
  end;
