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
  (cdif(f,h).1).x = Shift(f,h/2).x - Shift(f,-h/2).x
proof
  set f2 = Shift(f,-h/2);
  set f1 = Shift(f,h/2);
  (cdif(f,h).1).x = cdif(f,h).(0+1).x .= cD(cdif(f,h).0,h).x by Def8
    .= cD(f,h).x by Def8
    .= f.(x+h/2) - f.(x-h/2) by Th5
    .= f1.x - f.(x+-h/2) by Def2
    .= f1.x - f2.x by Def2;
  hence thesis;
end;
