 reserve n,s for Nat;

theorem Th72:
  (n + 1) choose 2 = n * (n + 1) / 2
  proof
    per cases;
    suppose
A1:   n <> 0; then
A2:   n = n -' 1 + 1 by XREAL_1:235,NAT_1:14;
A3:   n + 1 >= 1 + 1 by NAT_1:14,A1,XREAL_1:6;
      n -' 1 = n - 1 by XREAL_1:233,NAT_1:14,A1
            .= (n + 1) - 2; then
      (n + 1) choose 2 = ((n + 1)!) / (2! * ((n -' 1)!)) by NEWTON:def 3,A3
         .= (n! * (n + 1)) / (2 * ((n -' 1)!)) by NEWTON:15,14
         .= ((n -' 1)! * n * (n + 1)) / (2 * ((n -' 1)!)) by NEWTON:15,A2
         .= ((n -' 1)! * (n * (n + 1))) / (2 * ((n -' 1)!))
         .= (n * (n + 1)) / 2 by XCMPLX_1:91;
      hence thesis;
    end;
    suppose n = 0;
      hence thesis by NEWTON:def 3;
    end;
  end;
