reserve k, l, m, n, i, j for Nat,
  K, N for non empty Subset of NAT,
  Ke, Ne, Me for Subset of NAT,
  X,Y for set;
reserve f for Function of Segm n,Segm k;
reserve x,y for set;

theorem Th52:
  (n + 1) block n = (n + 1) choose 2
proof
  defpred P[Nat] means ($1+1) block $1 = ($1+1) choose 2;
A1: for k st P[k] holds P[k+1]
  proof
    let k such that
A2: P[k];
    set k1=k+1;
    thus (k1+1) block k1=k1*(k1 block k1)+(k1 block k) by Th46
      .=k1*1+(k1 choose 2) by A2,Th26
      .=k1+k1*(k1-1)/2 by Th51
      .=(k1+1)*(k1+1-1)/2
      .=(k1+1) choose 2 by Th51;
  end;
  1 block 0 =0 by Th31;
  then
A3: P[0] by NEWTON:def 3;
  for k holds P[k] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
