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 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;
    reconsider rfdk = fdif(r(#)f,h).(k+1) as Function of V,W by Th2;
    reconsider fdk = fdif(f,h).(k+1) as Function of V,W by Th2;
    fdif(r(#)f,h).(k+1+1)/.x = fD(fdif(r(#)f,h).(k+1),h)/.x by Def6
    .= rfdk/.(x+h) - rfdk/.x by Th3
    .= r * (fdk/.(x+h) - fdk/.x) by VECTSP_1:23,A3
    .= r * fD(fdk,h)/.x by Th3
    .= r * fdif(f,h).(k+1+1)/.x by Def6;
    hence thesis;
  end;
A6: X[0]
  proof
    let x;
    x in the carrier of V; then
A7: x in dom (r(#)f) by FUNCT_2:def 1;
    x + h in the carrier of V; 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,Def4X
    .= r * f/.(x+h) - r * f/.x by A7,Def4X
    .= r * (f/.(x+h) - f/.x) by VECTSP_1:23
    .= 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;
