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 Th7:
  fdif(r(#)f,h).(n+1).x = r* fdif(f,h).(n+1).x
proof
  defpred X[Nat] means
  for x holds fdif(r(#)f,h).($1+1).x = r* fdif(f,h).($1+1).x;
A1: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A2: for x holds fdif(r(#)f,h).(k+1).x = r* fdif(f,h).(k+1).x;
    let x;
A3: fdif(r(#)f,h).(k+1).x = r* fdif(f,h).(k+1).x & fdif(r(#)f,h).(k+1).(x+
    h) = r * fdif(f,h).(k+1).(x+h) by A2;
A4: fdif(r(#)f,h).(k+1) is Function of REAL,REAL by Th2;
A5: fdif(f,h).(k+1) is Function of REAL,REAL by Th2;
    fdif(r(#)f,h).(k+1+1).x = fD(fdif(r(#)f,h).(k+1),h).x by Def6
      .= r * fdif(f,h).(k+1).(x+h) - r * fdif(f,h).(k+1).x by A3,A4,Th3
      .= r * (fdif(f,h).(k+1).(x+h) - fdif(f,h).(k+1).x)
      .= r * fD(fdif(f,h).(k+1),h).x by A5,Th3
      .= r * fdif(f,h).(k+1+1).x by Def6;
    hence thesis;
  end;
A6: X[0]
  proof
    let x;
    x in REAL by XREAL_0:def 1; then
A7: x in dom (r(#)f) by FUNCT_2:def 1;
    x+h in REAL by XREAL_0:def 1; then
A8: x+h in dom (r(#)f) by FUNCT_2:def 1;
    fdif(r(#)f,h).(0+1).x = fD(fdif(r(#)f,h).0,h).x by Def6
      .= fD(r(#)f,h).x by Def6
      .= (r(#)f).(x+h) - (r(#)f).x by Th3
      .= r * f.(x+h) - (r(#)f).x by A8,VALUED_1:def 5
      .= r * f.(x+h) - r * f.x by A7,VALUED_1:def 5
      .= r * (f.(x+h) - f.x)
      .= r * fD(f,h).x by Th3
      .= r * fD(fdif(f,h).0,h).x by Def6
      .= r * fdif(f,h).(0+1).x by Def6;
    hence thesis;
  end;
  for n holds X[n] from NAT_1:sch 2(A6,A1);
  hence thesis;
end;
