reserve n for non zero Element of NAT;
reserve a,b,r,t for Real;
reserve
  Z for open Subset of REAL,
 y0 for VECTOR of REAL-NS n,
  G for Function of REAL-NS n,REAL-NS n;

theorem Th47:
 for n be Nat, g be Function of REAL,REAL
   st for x be Real holds g.x = ((x-a) |^ (n+1)) / ((n+1)! ) holds
   for x be Real holds g is_differentiable_in x
     & diff(g,x) = ((x-a) |^ n) / (n!)
proof
  let n be Nat;
  let g be Function of REAL,REAL;
A1: dom(g)=REAL by FUNCT_2:def 1;
  assume A2: for x be Real holds g.x = ((x-a) |^ (n+1)) / ((n+1)! );
  defpred X[set] means $1 in REAL;
  deffunc U(Real) = In(($1-a)|^ (n+1),REAL);
  consider f being PartFunc of REAL,REAL such that
A3: for d be Element of REAL holds d in dom f iff X[d] and
A4: for d be Element of REAL st d in dom f holds f/.d = U(d)
  from PARTFUN2:sch 2;
  for x be object st x in REAL holds x in dom f by A3; then
  REAL c=dom(f) by TARSKI:def 3; then
A5: dom(f)=REAL by XBOOLE_0:def 10;
A6: for d be Real holds f.d = (d-a) |^ (n+1)
  proof
    let d be Real;
A7:  d in REAL by XREAL_0:def 1;
    f/.d = In((d-a) |^ (n+1),REAL) by A4,A5,A7;
    hence thesis by A5,PARTFUN1:def 6,A7;
  end;
A8: f is Function of REAL,REAL by A5,FUNCT_2:67;
A9: dom ( ( 1 /( (n+1)!) ) (#) f ) = dom(f) by VALUED_1:def 5;
A10: now
    let x be Element of REAL;
    assume x in dom ((1/((n+1)!)) (#) f);
    hence (( 1/((n+1)!)) (#) f ).x = (1/((n+1)!)) * f.x by VALUED_1:def 5
      .= 1/((n+1)!)*( (x-a) |^ (n+1) ) by A6
      .= ((x-a) |^ (n+1))/((n+1)!) * 1 by XCMPLX_1:75
      .= g.x by A2;
  end;
A11: for x be Real holds (1/((n+1)!)) (#) f is_differentiable_in x
  & diff((1/((n+1)!)) (#) f,x) = ((x-a) |^ n)/(n!)
  proof
    let x be Real;
  A12: (n+1) / ((n+1)!) = (1*(n+1))/ (n! * (n+1)) by NEWTON:15
    .= 1/(n!) by XCMPLX_1:91;
  A13: f is_differentiable_in x by A8,A6,Th46;
    hence ( 1/((n+1)!) ) (#) f is_differentiable_in x by FDIFF_1:15;
    thus diff((1/((n+1)!)) (#) f,x) = (1/((n+1)!)) *diff(f,x) by A13,FDIFF_1:15
      .= diff(f,x)/((n+1)!)*1 by XCMPLX_1:75
      .=1 *( (n+1)* ((x-a) |^ n) )/ ((n+1)!) by A8,A6,Th46
      .=1*( (x-a) |^ n * ((n+1) / ((n+1)!)) ) by XCMPLX_1:74
      .=(x-a) |^ n / (n!) by A12,XCMPLX_1:99;
  end;
  ( (1/((n+1)!)) (#) f) =g by A10,PARTFUN1:5,A1,A5,A9;
  hence thesis by A11;
end;
