reserve C for non empty set;
reserve GF for Field,
        V for VectSp of GF,
        v,u for Element of V,
        W for Subset of V;
reserve f,f1,f2,f3 for PartFunc of C,V;
reserve F,G for Field,
        V for VectSp of F,
        W for VectSp of G;
reserve f,f1,f2 for Function of V, W;
reserve x,h for Element of V;
reserve r,r1,r2 for Element of G;
reserve n,m,k for Nat;

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-(2*1.F)"*h)
    = r * cdif(f,h).(k+1)/.(x-(2*1.F)"*h) &
    cdif(r(#)f,h).(k+1)/.(x+(2*1.F)"*h)
    = r * cdif(f,h).(k+1)/.(x+(2*1.F)"*h) by A2;
A4: cdif(r(#)f,h).(k+1) is Function of V,W by Th19;
A5: cdif(f,h).(k+1) is Function of V,W 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+(2*1.F)"*h)
      - cdif(r(#)f,h).(k+1)/.(x-(2*1.F)"*h) by A4,Th5
    .= r * (cdif(f,h).(k+1)/.(x+(2*1.F)"*h)
      - cdif(f,h).(k+1)/.(x-(2*1.F)"*h)) by VECTSP_1:23,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+(2*1.F)"*h in the carrier of V;
    then
A7: x+(2*1.F)"*h in dom (r(#)f) by FUNCT_2:def 1;
    x-(2*1.F)"*h in the carrier of V;
    then
A8: x-(2*1.F)"*h 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+(2*1.F)"*h) - (r(#)f)/.(x-(2*1.F)"*h) by Th5
    .= r * f/.(x+(2*1.F)"*h) - (r(#)f)/.(x-(2*1.F)"*h) by A7,Def4X
    .= r * f/.(x+(2*1.F)"*h) - r * f/.(x-(2*1.F)"*h) by A8,Def4X
    .= r * (f/.(x+(2*1.F)"*h) - f/.(x-(2*1.F)"*h)) by VECTSP_1:23
    .= 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;
