 reserve n,s for Nat;

theorem
  Triangle (n -' 1) + Triangle (n) = n^2
  proof
    per cases;
    suppose n <> 0; then
A1:   n -' 1 = n - 1 by XREAL_1:233,NAT_1:14;
      Triangle (n -' 1) =
           (n - 1) * (n - 1 + 1) / 2 by Th19,A1; then
      Triangle (n -' 1) + Triangle (n)
         = (n - 1) * (n + 1 - 1) / 2 + n * (n + 1) / 2 by Th19
        .= n ^2;
      hence thesis;
    end;
    suppose
A2:   n = 0; then
      Triangle (n -' 1) + Triangle (n) = Triangle (0 -' 1) + Triangle 0
        .= n^2 by A2,XREAL_0:def 2,Lm3;
      hence thesis;
    end;
  end;
