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 ((id REAL)^)`|Z = ((-1)(#)(( #Z 2)^))|Z
proof
  assume
A1: not 0 in Z;
  then (id REAL)^ is_differentiable_on Z by Lm2,Th43;
  then
A2: dom(((id REAL)^)`|Z) = Z by FDIFF_1:def 7;
A3: dom (( #Z 2)^) = REAL \ {0} by Th3;
  then
A4: Z c= dom (( #Z 2)^) by A1,ZFMISC_1:34;
A5: for x0 being Element of REAL st
    x0 in Z holds (((id REAL)^)`|Z).x0 = (((-1)(#)(( #Z 2)^))|Z). x0
  proof
A6: dom((-1)(#)(( #Z 2)^))=dom((( #Z (1+1))^)) by VALUED_1:def 5;
    let x0 be Element of REAL;
    assume
A7: x0 in Z;
    (id REAL)^ is_differentiable_on Z by A1,Lm2,Th43;
    then (((id REAL)^)`|Z).x0 = diff((id REAL)^,x0) by A7,FDIFF_1:def 7
      .=-1/x0^2 by A1,A7,Th45
      .=-1/(x0|^(1+1)) by NEWTON:81
      .=-1/x0 #Z 2 by PREPOWER:36
      .=-1/( #Z 2).x0 by TAYLOR_1:def 1
      .=-1*(( #Z 2).x0)" by XCMPLX_0:def 9
      .=-1*(( #Z 2)^).x0 by A4,A7,RFUNCT_1:def 2
      .=(-1)*(( #Z 2)^).x0
      .=((-1)(#)(( #Z 2)^)).x0 by A4,A7,A6,VALUED_1:def 5
      .=(((-1)(#)(( #Z 2)^))|Z).x0 by A7,FUNCT_1:49;
    hence thesis;
  end;
  dom(((-1)(#)(( #Z (1+1))^))|Z)=dom(((-1)(#)(( #Z 2)^)))/\Z by RELAT_1:61
    .=dom((( #Z 2)^))/\Z by VALUED_1:def 5
    .=Z by A1,A3,XBOOLE_1:28,ZFMISC_1:34;
  hence thesis by A2,A5,PARTFUN1:5;
end;
