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
  not 0 in Z & x0 in Z implies ((( #Z n)^)`|Z).x0 =- n/( #Z (n+1)).x0
proof
  assume that
A1: not 0 in Z and
A2: x0 in Z;
A3: ( #Z n)^ is_differentiable_on Z by A1,Th43;
  per cases;
  suppose
    n>0;
    then 0<0+n;
    then n>=1 by NAT_1:19;
    then reconsider i=n-1 as Element of NAT by INT_1:5;
    x0 #Z i<>0 by A1,A2,PREPOWER:38;
    then
A4: x0|^i<>0 by PREPOWER:36;
    ((( #Z n)^)`|Z).x0 = diff(( #Z n)^,x0) by A2,A3,FDIFF_1:def 7
      .=- (n* x0 #Z i) /( x0 #Z n)^2 by A1,A2,Th29
      .=- (n* x0 #Z i) /(x0 |^n)^2 by PREPOWER:36
      .= - (n* x0|^i) /( x0 |^n)^2 by PREPOWER:36
      .=- (n* x0|^i) /(( x0 |^(i+1))*( x0 |^1*x0 |^i)) by NEWTON:8
      .=- (n* x0|^i) /(( x0 |^(i+1)* x0 |^1)*x0 |^i)
      .=- (n* x0|^i) /( x0 |^(i+1+1)*x0 |^i) by NEWTON:8
      .=- n/x0 |^(i+2) by A4,XCMPLX_1:91
      .=- n/x0 #Z (i+2) by PREPOWER:36
      .=- n/( #Z (n+1)).x0 by TAYLOR_1:def 1;
    hence thesis;
  end;
  suppose
A5: n=0;
    ((( #Z n)^)`|Z).x0 = diff(( #Z n)^,x0) by A2,A3,FDIFF_1:def 7
      .=- (0* x0 #Z (n-1)) /( x0 #Z n)^2 by A1,A2,A5,Th29
      .=- n/( #Z (n+1)).x0 by A5;
    hence thesis;
  end;
end;
