reserve n for Nat,
  i for Integer,
  p, x, x0, y for Real,
  q for Rational,
  f for PartFunc of REAL,REAL;

theorem Th31:
  for f be PartFunc of REAL,REAL, Z be Subset of REAL, Z1 be open
Subset of REAL st Z1 c= Z for n be Nat st f is_differentiable_on n+1
  ,Z holds f is_differentiable_on n+1, Z1
proof
  let f be PartFunc of REAL,REAL;
  let Z be Subset of REAL;
  let Z1 be open Subset of REAL such that
A1: Z1 c= Z;
  let n be Nat such that
A2: f is_differentiable_on n+1, Z;
  now
    let k be Nat such that
A3: k <= (n+1)-1;
    (diff(f,Z).k) is_differentiable_on Z by A2,A3;
    then (diff(f,Z).k) is_differentiable_on Z1 by A1,FDIFF_1:26;
    then
A4: (diff(f,Z).k)|Z1 is_differentiable_on Z1 by FDIFF_2:16;
    n <= n+1 by NAT_1:11;
    then k <= n+1 by A3,XXREAL_0:2;
    hence (diff(f,Z1).k) is_differentiable_on Z1 by A1,A2,A4,Th23,Th30;
  end;
  hence thesis;
end;
