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 Th21:
  cdif(r(#)f,h).(n+1).x = r* cdif(f,h).(n+1).x
proof
  defpred X[Nat] means
for x holds cdif(r(#)f,h).($1+1).x = r* cdif(f,h).($1+1).x;
A1: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A2: for x holds cdif(r(#)f,h).(k+1).x = r* cdif(f,h).(k+1).x;
    let x;
A3: cdif(r(#)f,h).(k+1).(x-h/2) = r* cdif(f,h).(k+1).(x-h/2) & cdif(r(#)f,
    h).(k+ 1).(x+h/2) = r* cdif(f,h).(k+1).(x+h/2) by A2;
A4: cdif(r(#)f,h).(k+1) is Function of REAL,REAL by Th19;
A5: cdif(f,h).(k+1) is Function of REAL,REAL by Th19;
    cdif(r(#)f,h).(k+1+1).x = cD(cdif(r(#)f,h).(k+1),h).x by Def8
      .= cdif(r(#)f,h).(k+1).(x+h/2) - cdif(r(#)f,h).(k+1).(x-h/2) by A4,Th5
      .= r * (cdif(f,h).(k+1).(x+h/2) - cdif(f,h).(k+1).(x-h/2)) by A3
      .= r * cD(cdif(f,h).(k+1),h).x by A5,Th5
      .= r * cdif(f,h).(k+1+1).x by Def8;
    hence thesis;
  end;
A6: X[0]
  proof
    let x;
    x+h/2 in REAL by XREAL_0:def 1;
    then
A7: x+h/2 in dom (r(#)f) by FUNCT_2:def 1;
    x-h/2 in REAL by XREAL_0:def 1;
    then
A8: x-h/2 in dom (r(#)f) by FUNCT_2:def 1;
    cdif(r(#)f,h).(0+1).x = cD(cdif(r(#)f,h).0,h).x by Def8
      .= cD(r(#)f,h).x by Def8
      .= (r(#)f).(x+h/2) - (r(#)f).(x-h/2) by Th5
      .= r * f.(x+h/2) - (r(#)f).(x-h/2) by A7,VALUED_1:def 5
      .= r * f.(x+h/2) - r * f.(x-h/2) by A8,VALUED_1:def 5
      .= r * (f.(x+h/2) - f.(x-h/2))
      .= r * cD(f,h).x by Th5
      .= r * cD(cdif(f,h).0,h).x by Def8
      .= r * cdif(f,h).(0+1).x by Def8;
    hence thesis;
  end;
  for n holds X[n] from NAT_1:sch 2(A6,A1);
  hence thesis;
end;
