reserve x,r,a,x0,p for Real;
reserve n,i,m for Element of NAT;
reserve Z for open Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;
reserve k for Nat;

theorem
  n>=2 implies diff( #Z n,Z).3 = (n*(n-1)*(n-2)(#) #Z (n-3))|Z
proof
  assume
A1: 2<=n;
  then
A2: 2+(-1)<=n+0 by XREAL_1:7;
  reconsider m = n-2 as Element of NAT by A1,INT_1:5;
A3: #Z m is_differentiable_on Z by Th8,FDIFF_1:26;
  then
A4: (m+2)*(m+1)(#) #Z m is_differentiable_on Z by FDIFF_2:19;
  diff( #Z n,Z).3 = diff( #Z n,Z).(2+1)
    .= diff( #Z n,Z).2 `|Z by TAYLOR_1:def 5
    .= ((n*(n-1)(#) #Z (n-2))|Z) `|Z by A2,Th30
    .= ((m+2)*(m+1)(#) #Z m) `|Z by A4,FDIFF_2:16
    .= (m+2)*(m+1)(#)( #Z m`|Z ) by A3,FDIFF_2:19
    .= (m+2)*(m+1)(#)(( m(#) #Z(m-1))|Z) by Th28
    .=(n*(n-1)(#)((n-2)(#) #Z (n-3))) |Z by RFUNCT_1:49
    .=(n*(n-1)*(n-2)(#) #Z (n-3)) |Z by RFUNCT_1:17;
  hence thesis;
end;
