
theorem Th8:
  for n being Nat st n > 1 holds Catalan (n) = (2 * n -'
  2)! / ((n -' 1)! * (n!))
proof
  let n be Nat;
  assume
A1: n > 1;
  then
A2: 2 * 1 <= 2 * n by XREAL_1:64;
A3: n -' 1 + 1 = n by A1,XREAL_1:235;
A4: n -' 1 <= 2 * n -' 2 by A1,Th2;
  (2*n -' 2) - (n -' 1) = (2*n -' 2) - (n - 1) by A1,XREAL_1:233
    .= (2*n - 2) - (n - 1) by A2,XREAL_1:233
    .= n -' 1 by A1,XREAL_1:233;
  then
  ((2*n -' 2) choose (n -' 1)) = ((2*n -' 2)!) / ((n -' 1)! * ((n -' 1)!))
  by A4,NEWTON:def 3;
  then Catalan (n) = ((2*n -' 2)!) / ((n -' 1)! * ((n -' 1)!) * n) by
XCMPLX_1:78
    .= ((2*n -' 2)!) / ((n -' 1)! * (((n -' 1)!) * n))
    .= ((2*n -' 2)!) / ((n -' 1)! * (n!)) by A3,NEWTON:15;
  hence thesis;
end;
