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 Th14:
  bdif(r(#)f,h).(n+1).x = r* bdif(f,h).(n+1).x
proof
  defpred X[Nat] means
for x holds bdif(r(#)f,h).($1+1).x = r* bdif
  (f,h).($1+1).x;
A1: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A2: for x holds bdif(r(#)f,h).(k+1).x = r* bdif(f,h).(k+1).x;
    let x;
A3: bdif(r(#)f,h).(k+1).x = r* bdif(f,h).(k+1).x & bdif(r(#)f,h).(k+1).(x-
    h) = r * bdif(f,h).(k+1).(x-h) by A2;
A4: bdif(r(#)f,h).(k+1) is Function of REAL,REAL by Th12;
A5: bdif(f,h).(k+1) is Function of REAL,REAL by Th12;
    bdif(r(#)f,h).(k+1+1).x = bD(bdif(r(#)f,h).(k+1),h).x by Def7
      .= bdif(r(#)f,h).(k+1).x - bdif(r(#)f,h).(k+1).(x-h) by A4,Th4
      .= r * (bdif(f,h).(k+1).x - bdif(f,h).(k+1).(x-h)) by A3
      .= r * bD(bdif(f,h).(k+1),h).x by A5,Th4
      .= r * bdif(f,h).(k+1+1).x by Def7;
    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;
    bdif(r(#)f,h).(0+1).x = bD(bdif(r(#)f,h).0,h).x by Def7
      .= bD(r(#)f,h).x by Def7
      .= (r(#)f).x - (r(#)f).(x-h) by Th4
      .= (r(#)f).x - r * f.(x-h) by A8,VALUED_1:def 5
      .= r * f.x - r * f.(x-h) by A7,VALUED_1:def 5
      .= r * (f.x - f.(x-h))
      .= r * bD(f,h).x by Th4
      .= r * bD(bdif(f,h).0,h).x by Def7
      .= r * bdif(f,h).(0+1).x by Def7;
    hence thesis;
  end;
  for n holds X[n] from NAT_1:sch 2(A6,A1);
  hence thesis;
end;
