reserve a,b,i,j,k,l,m,n for Nat;

theorem
  (Newton_Coeff (2*n+1)).(n+1) = (Newton_Coeff (2*n+1)).(n+2)
proof
  reconsider k = 2*n+1 as Nat;
  A0: (n+1) + n >= (n+1) + 0 & n+(n+1) >= n + 0 by XREAL_1:6;
  A2: len (Newton_Coeff (2*n+1)) = (2*n + 1)+1 by NEWTON:def 5;
  1+(n+1) >= 1+0 & (n+2)+n >= (n+2)+0 by XREAL_1:6; then
  A4: n+2 in dom (Newton_Coeff (2*n+1)) & n+1 = (n+2)-1 by A2,FINSEQ_3:25;
  (2*n+1) - n = n+1; then
  reconsider l = k - n as Nat;
  (Newton_Coeff (2*n+1)).(n+1) = (2*n+1) choose n by NCI
  .= k choose l by A0,NEWTON:20
  .= (Newton_Coeff (2*n+1)).(n+2) by A4,NEWTON:def 5;
  hence thesis;
end;
