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(f1-f2,h).(n+1).x = cdif(f1,h).(n+1).x - cdif(f2,h).(n+1).x
proof
  defpred X[Nat] means
for x holds cdif(f1-f2,h).($1+1).x = cdif(f1
  ,h).($1+1).x - cdif(f2,h).($1+1).x;
A1: X[0]
  proof
    let x;
    x-h/2 in REAL by XREAL_0:def 1;
    then x-h/2 in dom f1 & x-h/2 in dom f2 by FUNCT_2:def 1;
    then x-h/2 in dom f1 /\ dom f2 by XBOOLE_0:def 4;
    then
A2: x-h/2 in dom (f1-f2) by VALUED_1:12;
    x+h/2 in REAL by XREAL_0:def 1;
    then x+h/2 in dom f1 & x+h/2 in dom f2 by FUNCT_2:def 1;
    then x+h/2 in dom f1 /\ dom f2 by XBOOLE_0:def 4;
    then
A3: x+h/2 in dom (f1-f2) by VALUED_1:12;
    cdif(f1-f2,h).(0+1).x = cD(cdif(f1-f2,h).0,h).x by Def8
      .= cD(f1-f2,h).x by Def8
      .= (f1-f2).(x+h/2) - (f1-f2).(x-h/2) by Th5
      .= f1.(x+h/2) - f2.(x+h/2) - (f1-f2).(x-h/2) by A3,VALUED_1:13
      .= f1.(x+h/2) - f2.(x+h/2) - (f1.(x-h/2) - f2.(x-h/2)) by A2,VALUED_1:13
      .= (f1.(x+h/2) - f1.(x-h/2)) - (f2.(x+h/2) - f2.(x-h/2))
      .= cD(f1,h).x - (f2.(x+h/2) - f2.(x-h/2)) by Th5
      .= cD(f1,h).x - cD(f2,h).x by Th5
      .= cD(cdif(f1,h).0,h).x - cD(f2,h).x by Def8
      .= cD(cdif(f1,h).0,h).x - cD(cdif(f2,h).0,h).x by Def8
      .= cdif(f1,h).(0+1).x - cD(cdif(f2,h).0,h).x by Def8
      .= cdif(f1,h).(0+1).x - cdif(f2,h).(0+1).x by Def8;
    hence thesis;
  end;
A4: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A5: for x holds cdif(f1-f2,h).(k+1).x = cdif(f1,h).(k+1).x - cdif(f2,
    h).(k+1).x;
    let x;
A6: cdif(f1-f2,h).(k+1).(x-h/2) = cdif(f1,h).(k+1).(x-h/2) - cdif(f2,h).(
k+1).(x- h/2) & cdif(f1-f2,h).(k+1).(x+h/2) = cdif(f1,h).(k+1).(x+h/2) - cdif(
    f2,h).(k+ 1).(x+h/2) by A5;
A7: cdif(f1-f2,h).(k+1) is Function of REAL,REAL by Th19;
A8: cdif(f2,h).(k+1) is Function of REAL,REAL by Th19;
A9: cdif(f1,h).(k+1) is Function of REAL,REAL by Th19;
    cdif(f1-f2,h).(k+1+1).x = cD(cdif(f1-f2,h).(k+1),h).x by Def8
      .= cdif(f1-f2,h).(k+1).(x+h/2) - cdif(f1-f2,h).(k+1).(x-h/2) by A7,Th5
      .= (cdif(f1,h).(k+1).(x+h/2) - cdif(f1,h).(k+1).(x-h/2)) - (cdif(f2,h)
    .(k+1).(x+h/2) - cdif(f2,h).(k+1).(x-h/2)) by A6
      .= cD(cdif(f1,h).(k+1),h).x - (cdif(f2,h).(k+1).(x+h/2) - cdif(f2,h).(
    k+1).(x-h/2)) by A9,Th5
      .= cD(cdif(f1,h).(k+1),h).x - cD(cdif(f2,h).(k+1),h).x by A8,Th5
      .= cdif(f1,h).(k+1+1).x - cD(cdif(f2,h).(k+1),h).x by Def8
      .= cdif(f1,h).(k+1+1).x - cdif(f2,h).(k+1+1).x by Def8;
    hence thesis;
  end;
  for n holds X[n] from NAT_1:sch 2(A1,A4);
  hence thesis;
end;
