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
  (fdif(f,h).1).x = Shift(f,h).x - f.x
proof
  set f1 = Shift(f,h);
  (fdif(f,h).1).x = fdif(f,h).(0+1).x .= fD(fdif(f,h).0,h).x by Def6
    .= fD(f,h).x by Def6
    .= f.(x+h) - f.x by Th3
    .= f1.x - f.x by Def2;
  hence thesis;
end;
