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
  fdif(f1-f2,h).(n+1)/.x = fdif(f1,h).(n+1)/.x - fdif(f2,h).(n+1)/.x
proof
  defpred X[Nat] means
  for x holds fdif(f1-f2,h).($1+1)/.x = fdif(f1,h).($1+1)/.x
  - fdif(f2,h).($1+1)/.x;
A1: X[0]
  proof
    let x;
    x in the carrier of V;
    then x in dom f1 & x in dom f2 by FUNCT_2:def 1;
    then x in dom f1 /\ dom f2 by XBOOLE_0:def 4; then
A2: x in dom (f1-f2) by VFUNCT_1:def 2;
    x + h in the carrier of V;
    then x + h in dom f1 & x + h in dom f2 by FUNCT_2:def 1;
    then x + h in dom f1 /\ dom f2 by XBOOLE_0:def 4; then
A3: x + h in dom (f1-f2) by VFUNCT_1:def 2;
    fdif(f1-f2,h).(0+1)/.x = fD(fdif(f1-f2,h).0,h)/.x by Def6
    .= fD(f1-f2,h)/.x by Def6
    .= (f1-f2)/.(x+h) - (f1-f2)/.x by Th3
    .= f1/.(x+h) - f2/.(x+h) - (f1-f2)/.x by A3,VFUNCT_1:def 2
    .= f1/.(x+h) - f2/.(x+h) - (f1/.x - f2/.x) by A2,VFUNCT_1:def 2
    .= (f1/.(x+h) - f2/.(x+h) - f1/.x) + f2/.x by RLVECT_1:29
    .= (f1/.(x+h) - (f1/.x + f2/.(x+h))) + f2/.x by RLVECT_1:27
    .= ((f1/.(x+h) - f1/.x) - f2/.(x+h)) + f2/.x by RLVECT_1:27
    .= ((fD(f1,h)/.x) - f2/.(x+h)) + f2/.x by Th3
    .= fD(f1,h).x - (f2/.(x+h) - f2/.x) by RLVECT_1:29
    .= fD(f1,h)/.x - fD(f2,h)/.x by Th3
    .= fD(fdif(f1,h).0,h)/.x - fD(f2,h)/.x by Def6
    .= fD(fdif(f1,h).0,h)/.x - fD(fdif(f2,h).0,h)/.x by Def6
    .= fdif(f1,h).(0+1)/.x - fD(fdif(f2,h).0,h)/.x by Def6
    .= fdif(f1,h).(0+1)/.x - fdif(f2,h).(0+1)/.x by Def6;
    hence thesis;
  end;
A4: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A5: for x holds
    fdif(f1-f2,h).(k+1)/.x = fdif(f1,h).(k+1)/.x - fdif(f2,h).(k+1)/.x;
    let x;
A6: fdif(f1-f2,h).(k+1)/.x = fdif(f1,h).(k+1)/.x - fdif(f2,h).(k+1)/.x &
    fdif(f1-f2,h).(k+1)/.(x+h)
    = fdif(f1,h).(k+1)/.(x+h) - fdif(f2,h).(k+1)/.(x+h) by A5;
    reconsider fd12k1 = fdif(f1-f2,h).(k+1) as Function of V, W by Th2;
    reconsider fd2k = fdif(f2,h).(k+1) as Function of V,W by Th2;
    reconsider fd1k = fdif(f1,h).(k+1) as Function of V, W by Th2;
    reconsider fdiff12 = fdif(f1-f2,h).(k+1) as Function of V, W by Th2;
    reconsider fdiff2 = fdif(f2,h).(k+1) as Function of V, W by Th2;
    reconsider fdiff1 = fdif(f1,h).(k+1) as Function of V, W by Th2;
A12: fD(fdif(f1,h).(k+1),h)/.x = fD(fdiff1,h)/.x
    .= fdif(f1,h).(k+1)/.(x+h) - fdif(f1,h).(k+1)/.x by Th3;
A13: fD(fdif(f2,h).(k+1),h)/.x = fD(fdiff2,h)/.x
    .= fdif(f2,h).(k+1)/.(x+h) - fdif(f2,h).(k+1)/.x by Th3;
    fdif(f1-f2,h).(k+1+1)/.x = fD(fdif(f1-f2,h).(k+1),h)/.x by Def6
    .= fd12k1/.(x+h) - fd12k1/.x by Th3
    .= (fdif(f1,h).(k+1)/.(x+h) + (-fdif(f2,h).(k+1)/.(x+h))
      - fdif(f1,h).(k+1)/.x) + fdif(f2,h).(k+1)/.x by RLVECT_1:29,A6
    .= (fdif(f1,h).(k+1)/.(x+h) + ((-fdif(f2,h).(k+1)/.(x+h))
      +(-fdif(f1,h).(k+1)/.x))) + fdif(f2,h).(k+1)/.x by RLVECT_1:def 3
    .= (fdif(f1,h).(k+1)/.(x+h) + ((-fdif(f1,h).(k+1)/.x))
      - fdif(f2,h).(k+1)/.(x+h)) + fdif(f2,h).(k+1)/.x by RLVECT_1:def 3
    .= fD(fdif(f1,h).(k+1),h)/.x - (fdif(f2,h).(k+1)/.(x+h)
      - fdif(f2,h).(k+1)/.x) by A12,RLVECT_1:29
    .= fdif(f1,h).(k+1+1)/.x - fD(fdif(f2,h).(k+1),h)/.x by Def6,A13
    .= fdif(f1,h).(k+1+1)/.x - fdif(f2,h).(k+1+1)/.x by Def6;
    hence thesis;
  end;
  for n holds X[n] from NAT_1:sch 2(A1,A4);
  hence thesis;
end;
