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 Th4:
  bD(f,h).x = f.x - f.(x-h)
proof
  thus bD(f,h).x = (-fD(f,-h)).x by RFUNCT_1:19
    .= - fD(f,-h).x by VALUED_1:8
    .= - (f.(x+-h) - f.x) by Th3
    .= f.x - f.(x-h);
end;
