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
  f is constant implies for x holds cdif(f,h).(n+1).x=0
proof
  defpred X[Nat] means for x holds cdif(f,h).($1+1).x=0;
  assume
A1: f is constant;
A2: for x holds f.(x+h/2) - f.(x-h/2) = 0
  proof
    let x;
    x-h/2 in REAL by XREAL_0:def 1;
    then
A3: x-h/2 in dom f by FUNCT_2:def 1;
    x+h/2 in REAL by XREAL_0:def 1;
    then x+h/2 in dom f by FUNCT_2:def 1;
    then f.(x+h/2) = f.(x-h/2) by A1,A3,FUNCT_1:def 10;
    hence thesis;
  end;
A4: X[0]
  proof
    let x;
    thus 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
      .= 0 by A2;
  end;
A5: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A6: for x holds cdif(f,h).(k+1).x=0;
    let x;
A7: cdif(f,h).(k+1).(x-h/2)=0 by A6;
A8: cdif(f,h).(k+1) is Function of REAL,REAL by Th19;
    (cdif(f,h).(k+2)).x = (cdif(f,h).(k+1+1)).x
      .= cD(cdif(f,h).(k+1),h).x by Def8
      .= cdif(f,h).(k+1).(x+h/2) - cdif(f,h).(k+1).(x-h/2) by A8,Th5
      .= 0 by A6,A7;
    hence thesis;
  end;
  for n holds X[n] from NAT_1:sch 2(A4,A5);
  hence thesis;
end;
