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;
reserve S for Seq_Sequence;

theorem
  for f being PartFunc of REAL, REAL st x+h/2 in dom f & x-h/2 in dom f holds
  cD(f,h).x = f.(x+h/2) - f.(x-h/2)
proof
  let f be PartFunc of REAL, REAL;
  assume
A1: x+h/2 in dom f & x-h/2 in dom f;
A2: dom Shift(f,-h/2) = - -(h/2) ++ dom f by Def1;
A3: dom Shift(f,h/2) = - (h/2) ++ dom f by Def1;
A4: -(h/2) + (x + (h/2)) in (-(h/2) ++ dom f) by A1,MEASURE6:46; then
A5: Shift(f,h/2).x = f.(x+h/2) by Def1;
A6: h/2 + (x - (h/2)) in (- -(h/2) ++ dom f) by A1,MEASURE6:46; then
A7: Shift(f,-h/2).x = f.(x+-h/2) by Def1;
  x in (dom Shift(f,h/2)) /\ (dom Shift(f,-h/2))
    by A4,A6,A3,A2,XBOOLE_0:def 4; then
  x in dom cD(f,h) by VALUED_1:12;
  hence thesis by A7,A5,VALUED_1:13;
end;
