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
  n>=1 & f1 is_differentiable_on n,Z implies diff(f1,Z).1=f1`|Z
proof
  assume n>=1 & f1 is_differentiable_on n,Z;
  then
A1: f1 is_differentiable_on Z by Th7;
  diff(f1,Z).1 = diff(f1,Z).(1+0) .=diff(f1,Z).0`|Z by TAYLOR_1:def 5
    .=f1|Z`|Z by TAYLOR_1:def 5
    .=f1`|Z by A1,FDIFF_2:16;
  hence thesis;
end;
