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 Th47:
  x <> 0 implies ( #Z 2)^ is_differentiable_in x & diff(( #Z 2)^,x
  ) = -(2 * x)/(x #Z 2)^2
proof
A1: ( #Z 2).x = x #Z 2 & x #Z 2 = x|^2 by PREPOWER:36,TAYLOR_1:def 1;
  assume x<>0;
  then
A2: ( #Z 2).x <>0 by A1,PREPOWER:5;
A3: #Z 2 is_differentiable_in x by TAYLOR_1:2;
  then diff(( #Z 2)^,x) = - diff( #Z 2,x)/(( #Z 2).x)^2 by A2,FDIFF_2:15
    .= -(2 * x #Z (2-1))/(( #Z 2).x)^2 by TAYLOR_1:2
    .= -(2 * x #Z 1)/(x #Z 2)^2 by TAYLOR_1:def 1
    .= -(2 * x)/(x #Z 2)^2 by PREPOWER:35;
  hence thesis by A2,A3,FDIFF_2:15;
end;
