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 Th45: :: FDIFF_4:14
  x <> 0 implies (id REAL)^ is_differentiable_in x & diff((id REAL
  )^,x) = -1/x^2
proof
  set f = id REAL;
  assume
A1: x<>0;
   reconsider xx=x as Element of REAL by XREAL_0:def 1;
  f.x = x #Z 1 & x #Z 1 = x|^1 by Lm2,PREPOWER:36,TAYLOR_1:def 1;
  then
A2: f.x <> 0 by A1;
A3: f is_differentiable_in x by Lm2,TAYLOR_1:2;
  then diff(f^,x) = - diff(f,x)/(f.x)^2 by A2,FDIFF_2:15
    .=-(1 * x #Z (1-1))/(f.xx)^2 by Lm2,TAYLOR_1:2
    .=-(1 * x #Z 0)/x^2
    .=-1/x^2 by PREPOWER:34;
  hence thesis by A2,A3,FDIFF_2:15;
end;
