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 in dom f & x-h in dom f holds
  bD(f,h).x = f.x - f.(x-h)
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 bD(f,h) by VALUED_1:12;
  hence thesis by A4,VALUED_1:13;
end;
