reserve n,m,k,i for Nat,
  h,r,r1,r2,x0,x1,x2,x for Real,
  S for Functional_Sequence of REAL,REAL,
  y for set;
reserve f,f1,f2 for Function of REAL,REAL;

theorem
  for f being PartFunc of REAL, REAL st x in dom f & x+h in dom f holds
    fD(f,h).x = f.(x+h) - f.x
proof
  let f be PartFunc of REAL, REAL;
  assume
A1: x in dom f & x+h in dom f;
A2: dom Shift(f,h) = -h ++ dom f by Def1;
A3: (-h) + (x + h) in ((-h) ++ dom f) by A1,MEASURE6:46; then
A4: Shift(f,h).x = f.(x+h) by Def1;
  x in (dom Shift(f,h)) /\ dom f by A3,A2,A1,XBOOLE_0:def 4; then
  x in dom fD(f,h) by VALUED_1:12;
  hence fD(f,h).x = f.(x+h) - f.x by A4,VALUED_1:13;
end;
