reserve y for set;
reserve x,a,b,c for Real;
reserve n for Element of NAT;
reserve Z for open Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem Th14:
  Z c= dom f & (for x st x in Z holds f.x = a+x & f.x<>0) implies
  f^ is_differentiable_on Z & for x st x in Z holds ( (f^)`|Z).x= -1/(a+x)^2
proof
  assume that
A1: Z c= dom f and
A2: for x st x in Z holds f.x = a+x & f.x<>0;
A3: for x st x in Z holds f.x = 1*x+a by A2;
  then
A4: f is_differentiable_on Z by A1,FDIFF_1:23;
A5: for x st x in Z holds f.x<>0 by A2;
  then
A6: f^ is_differentiable_on Z by A4,FDIFF_2:22;
  now
    let x;
    assume
A7: x in Z;
    then
A8: f.x<>0 & f is_differentiable_in x by A2,A4,FDIFF_1:9;
    ( (f^)`|Z).x= diff(f^,x) by A6,A7,FDIFF_1:def 7
      .= - diff(f,x)/( f.x)^2 by A8,FDIFF_2:15
      .=- (f`|Z).x/( f.x)^2 by A4,A7,FDIFF_1:def 7
      .=-1/( f.x)^2 by A1,A3,A7,FDIFF_1:23
      .=-1/(a+x)^2 by A2,A7;
    hence ( (f^)`|Z).x=-1/(a+x)^2;
  end;
  hence thesis by A4,A5,FDIFF_2:22;
end;
