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

theorem Th23:
  for f be PartFunc of REAL,REAL, Z be Subset of REAL, n be
Nat st f is_differentiable_on n,Z for m be Nat st m <= n
  holds f is_differentiable_on m, Z
proof
  let f be PartFunc of REAL,REAL;
  let Z be Subset of REAL;
  let n be Nat such that
A1: f is_differentiable_on n, Z;
  let m be Nat such that
A2: m <=n;
  now
A3: m-1 <=n-1 by A2,XREAL_1:13;
    let i be Nat;
    assume i <=m-1;
    then i <=n-1 by A3,XXREAL_0:2;
    hence diff(f,Z).i is_differentiable_on Z by A1;
  end;
  hence thesis;
end;
