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
  for n,m be Element of NAT,r,x be Element of REAL st n>m & r>0
    holds Maclaurin( #Z
  n,].-r,r.[,x).m = 0 & Maclaurin( #Z n,].-r,r.[,x).n = x|^ n
proof
  let n,m be Element of NAT;
  let r,x be Element of REAL;
  assume that
A1: n>m and
A2: r>0;
  |.0-0.|=0 by ABSVALUE:2;
  then
A3: 0 in ].0-r,0+r.[ by A2,RCOMP_1:1;
  reconsider s = n-m as Element of NAT by A1,INT_1:5;
A4: n-m>m-m by A1,XREAL_1:9;
A5: Maclaurin( #Z n,].-r,r.[,x).n =Taylor( #Z n,].-r,r.[,0,x).n by
TAYLOR_2:def 1
    .=(x-0)|^ n by A1,A3,Th41
    .=x|^ n;
  Maclaurin( #Z n,].-r,r.[,x).m =Taylor( #Z n,].-r,r.[,0,x).m by TAYLOR_2:def 1
    .=(n choose m)*0 #Z (n-m)*(x-0)|^ m by A1,A3,Th41
    .=(n choose m)*(0|^s)*x|^ m by PREPOWER:36
    .=(n choose m)*0*x|^ m by A4,NEWTON:84
    .=0;
  hence thesis by A5;
end;
