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(f1-f2,h).(n+1).x = fdif(f1,h).(n+1).x - fdif(f2,h).(n+1).x
proof
  defpred X[Nat] means
for x holds fdif(f1-f2,h).($1+1).x = fdif(f1
  ,h).($1+1).x - fdif(f2,h).($1+1).x;
A1: X[0]
  proof
    let x;
    x in REAL by XREAL_0:def 1;
    then x in dom f1 & x in dom f2 by FUNCT_2:def 1;
    then x in dom f1 /\ dom f2 by XBOOLE_0:def 4;
    then
A2: x in dom (f1-f2) by VALUED_1:12;
    x+h in REAL by XREAL_0:def 1;
    then x+h in dom f1 & x+h in dom f2 by FUNCT_2:def 1;
    then x+h in dom f1 /\ dom f2 by XBOOLE_0:def 4;
    then
A3: x+h in dom (f1-f2) by VALUED_1:12;
    fdif(f1-f2,h).(0+1).x = fD(fdif(f1-f2,h).0,h).x by Def6
      .= fD(f1-f2,h).x by Def6
      .= (f1-f2).(x+h) - (f1-f2).x by Th3
      .= f1.(x+h) - f2.(x+h) - (f1-f2).x by A3,VALUED_1:13
      .= f1.(x+h) - f2.(x+h) - (f1.x - f2.x) by A2,VALUED_1:13
      .= (f1.(x+h) - f1.x) - (f2.(x+h) - f2.x)
      .= fD(f1,h).x - (f2.(x+h) - f2.x) by Th3
      .= fD(f1,h).x - fD(f2,h).x by Th3
      .= fD(fdif(f1,h).0,h).x - fD(f2,h).x by Def6
      .= fD(fdif(f1,h).0,h).x - fD(fdif(f2,h).0,h).x by Def6
      .= fdif(f1,h).(0+1).x - fD(fdif(f2,h).0,h).x by Def6
      .= fdif(f1,h).(0+1).x - fdif(f2,h).(0+1).x by Def6;
    hence thesis;
  end;
A4: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A5: for x holds fdif(f1-f2,h).(k+1).x = fdif(f1,h).(k+1).x - fdif(f2,
    h).(k+1).x;
    let x;
A6: fdif(f1-f2,h).(k+1).x = fdif(f1,h).(k+1).x - fdif(f2,h).(k+1).x &
fdif(f1-f2,h).(k+1).(x+h) = fdif(f1,h).(k+1).(x+h) - fdif(f2,h).(k+1).(x+h) by
A5;
A7: fdif(f1-f2,h).(k+1) is Function of REAL,REAL by Th2;
A8: fdif(f2,h).(k+1) is Function of REAL,REAL by Th2;
A9: fdif(f1,h).(k+1) is Function of REAL,REAL by Th2;
    fdif(f1-f2,h).(k+1+1).x = fD(fdif(f1-f2,h).(k+1),h).x by Def6
      .= fdif(f1-f2,h).(k+1).(x+h) - fdif(f1-f2,h).(k+1).x by A7,Th3
      .= (fdif(f1,h).(k+1).(x+h) - fdif(f1,h).(k+1).x) - (fdif(f2,h).(k+1).(
    x+h) - fdif(f2,h).(k+1).x) by A6
      .= fD(fdif(f1,h).(k+1),h).x - (fdif(f2,h).(k+1).(x+h) - fdif(f2,h).(k+
    1).x) by A9,Th3
      .= fD(fdif(f1,h).(k+1),h).x - fD(fdif(f2,h).(k+1),h).x by A8,Th3
      .= fdif(f1,h).(k+1+1).x - fD(fdif(f2,h).(k+1),h).x by Def6
      .= fdif(f1,h).(k+1+1).x - fdif(f2,h).(k+1+1).x by Def6;
    hence thesis;
  end;
  for n holds X[n] from NAT_1:sch 2(A1,A4);
  hence thesis;
end;
