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 Th41:
  x0 in Z & n>m implies Taylor( #Z n,Z,x0,x).m = (n choose m)*x0
  #Z (n-m) *(x-x0)|^ m & Taylor( #Z n,Z,x0,x).n = (x-x0)|^ n
proof
  assume that
A1: x0 in Z and
A2: n>m;
A3: Taylor( #Z n,Z,x0,x).n =(diff( #Z n,Z).n).x0*(x-x0)|^ n / (n!) by
TAYLOR_1:def 7
    .=(x-x0)|^ n*(n!)/ (n!) by A1,Th37
    .=(x-x0)|^ n by XCMPLX_1:89;
  Taylor( #Z n,Z,x0,x).m = (diff( #Z n,Z).m).x0*(x-x0)|^ m / (m!) by
TAYLOR_1:def 7
    .= (n choose m)*(m!)*x0 #Z (n-m)*(x-x0)|^ m / (m!) by A1,A2,Th36
    .=(n choose m)*(x0 #Z (n-m)*(x-x0)|^ m)*(m!)/ (m!)
    .= (n choose m)*x0 #Z (n-m)*(x-x0)|^ m by XCMPLX_1:89;
  hence thesis by A3;
end;
