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 Th6:
  f|Z is_differentiable_on Z implies f is_differentiable_on Z
proof
A1: dom (f|Z) c= dom f by RELAT_1:60;
  assume
A2: f|Z is_differentiable_on Z;
  then
A3: for x st x in Z holds f|Z is_differentiable_in x by FDIFF_1:9;
  Z c= dom (f|Z) by A2,FDIFF_1:9;
  then Z c= dom f by A1;
  hence thesis by A3,FDIFF_1:def 6;
end;
