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 implies (( #Z 2)^)`|Z = ((-2)(#)(( #Z 3)^))|Z
proof
  assume
A1: not 0 in Z;
  then ( #Z 2)^ is_differentiable_on Z by Th43;
  then
A2: dom((( #Z 2)^)`|Z) = Z by FDIFF_1:def 7;
  Z c= REAL \ {0} by A1,ZFMISC_1:34;
  then
A3: Z c= dom (( #Z 3)^) by Th3;
A4: for x0 being Element of REAL
    st x0 in Z holds ((( #Z 2)^)`|Z).x0 = (((-2)(#)(( #Z 3)^))|Z).x0
  proof
    reconsider i=2-1 as Element of NAT;
    let x0 be Element of REAL;
A5: dom((-2)(#)(( #Z 3)^))=dom((( #Z 3)^)) by VALUED_1:def 5;
    assume
A6: x0 in Z;
    then x0 #Z i<>0 by A1,PREPOWER:38;
    then
A7: x0|^i<>0 by PREPOWER:36;
    ( #Z 2)^ is_differentiable_on Z by A1,Th43;
    then ((( #Z 2)^)`|Z).x0 = diff(( #Z 2)^,x0) by A6,FDIFF_1:def 7
      .= - (2* x0)/( x0 #Z 2)^2 by A1,A6,Th47
      .= - (2* x0 #Z 1) /( x0 #Z 2)^2 by PREPOWER:35
      .=- (2* x0 #Z 1) /(x0 |^2)^2 by PREPOWER:36
      .=- (2* x0|^1) /(( x0 |^2)*( x0 |^(1+1))) by PREPOWER:36
      .=- (2* x0|^1) /(( x0 |^2)*( x0 |^1*x0 |^1)) by NEWTON:8
      .=- (2* x0|^1) /(( x0 |^(2)* x0 |^1)*x0 |^1)
      .=- (2* x0|^1) /( x0 |^(2+1)*x0 |^1) by NEWTON:8
      .=- 2/x0 |^(1+2) by A7,XCMPLX_1:91
      .=- 2/x0 #Z 3 by PREPOWER:36
      .=- 2/( #Z 3).x0 by TAYLOR_1:def 1
      .=-2*(( #Z 3).x0)" by XCMPLX_0:def 9
      .=-2*(( #Z 3)^).x0 by A3,A6,RFUNCT_1:def 2
      .=(-2)*(( #Z 3)^).x0
      .=((-2)(#)(( #Z 3)^)).x0 by A3,A6,A5,VALUED_1:def 5
      .=(((-2)(#)(( #Z 3)^))|Z).x0 by A6,FUNCT_1:49;
    hence thesis;
  end;
  dom(((-2)(#)(( #Z 3)^))|Z) = dom(((-2)(#)(( #Z 3)^)))/\Z by RELAT_1:61
    .=dom(( #Z 3)^)/\Z by VALUED_1:def 5
    .=Z by A3,XBOOLE_1:28;
  hence thesis by A2,A4,PARTFUN1:5;
end;
