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
  (bdif(f,h).1).x = f.x - Shift(f,-h).x
proof
  set f2 = Shift(f,-h);
  (bdif(f,h).1).x = bdif(f,h).(0+1).x .= bD(bdif(f,h).0,h).x by Def7
    .= bD(f,h).x by Def7
    .= f.x - f.(x-h) by Th4
    .= f.x - f.(x+-h)
    .= f.x - f2.x by Def2;
  hence thesis;
end;
