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
  f is_differentiable_on Z implies diff(f,Z).1 = f`|Z
proof
  assume
A1: f is_differentiable_on Z;
  then
A2: dom(f`|Z) = Z by FDIFF_1:def 7;
A3: for x being Element of REAL st x in Z holds (diff(f,Z).1).x = (f`|Z).x
  proof
    let x be Element of REAL;
    assume x in Z;
    (diff(f,Z).1).x =(diff(f,Z).(1+0)).x
      .=((diff(f,Z).0)`|Z).x by TAYLOR_1:def 5
      .=(f|Z`|Z).x by TAYLOR_1:def 5
      .=(f`|Z).x by A1,FDIFF_2:16;
    hence thesis;
  end;
  dom(diff(f,Z).1) = dom(diff(f,Z).(1+0))
    .= dom((diff(f,Z).0)`|Z) by TAYLOR_1:def 5
    .= dom(f|Z`|Z) by TAYLOR_1:def 5
    .= dom(f`|Z) by A1,FDIFF_2:16
    .= Z by A1,FDIFF_1:def 7;
  hence thesis by A2,A3,PARTFUN1:5;
end;
