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
  cdif(f1-f2,h).(n+1)/.x = cdif(f1,h).(n+1)/.x - cdif(f2,h).(n+1)/.x
proof
  defpred X[Nat] means
  for x holds cdif(f1-f2,h).($1+1)/.x = cdif(f1,h).($1+1)/.x
  - cdif(f2,h).($1+1)/.x;
A1: X[0]
  proof
    let x;
    x-(2*1.F)"*h in the carrier of V;
    then x-(2*1.F)"*h in dom f1 & x-(2*1.F)"*h in dom f2 by FUNCT_2:def 1;
    then x-(2*1.F)"*h in dom f1 /\ dom f2 by XBOOLE_0:def 4;
    then
A2: x-(2*1.F)"*h in dom (f1-f2) by VFUNCT_1:def 2;
    x+(2*1.F)"*h in the carrier of V;
    then x+(2*1.F)"*h in dom f1 & x+(2*1.F)"*h in dom f2 by FUNCT_2:def 1;
    then x+(2*1.F)"*h in dom f1 /\ dom f2 by XBOOLE_0:def 4;
    then
A3: x+(2*1.F)"*h in dom (f1-f2) by VFUNCT_1:def 2;
    cdif(f1-f2,h).(0+1)/.x = cD(cdif(f1-f2,h).0,h)/.x by Def8
    .= cD(f1-f2,h)/.x by Def8
    .= (f1-f2)/.(x+(2*1.F)"*h) - (f1-f2)/.(x-(2*1.F)"*h) by Th5
    .= f1/.(x+(2*1.F)"*h) - f2/.(x+(2*1.F)"*h) - (f1-f2)/.(x-(2*1.F)"*h)
    by A3,VFUNCT_1:def 2
    .= (f1/.(x+(2*1.F)"*h) - f2/.(x+(2*1.F)"*h)) - (f1/.(x-(2*1.F)"*h)
      - f2/.(x-(2*1.F)"*h)) by A2,VFUNCT_1:def 2
    .= ((f1/.(x+(2*1.F)"*h) - f2/.(x+(2*1.F)"*h)) - f1/.(x-(2*1.F)"*h))
      + f2/.(x-(2*1.F)"*h) by RLVECT_1:29
    .= (f1/.(x+(2*1.F)"*h) - (f1/.(x-(2*1.F)"*h) + f2/.(x+(2*1.F)"*h)))
      + f2/.(x-(2*1.F)"*h) by RLVECT_1:27
    .= f1/.(x+(2*1.F)"*h) - ((f1/.(x-(2*1.F)"*h) + f2/.(x+(2*1.F)"*h))
      - f2/.(x-(2*1.F)"*h)) by RLVECT_1:29
    .= f1/.(x+(2*1.F)"*h) - (f1/.(x-(2*1.F)"*h) + (f2/.(x+(2*1.F)"*h)
      - f2/.(x-(2*1.F)"*h))) by RLVECT_1:28
    .= (f1/.(x+(2*1.F)"*h) - f1/.(x-(2*1.F)"*h)) - (f2/.(x+(2*1.F)"*h)
      - f2/.(x-(2*1.F)"*h)) by RLVECT_1:27
    .= cD(f1,h)/.x - (f2/.(x+(2*1.F)"*h) - f2/.(x-(2*1.F)"*h)) by Th5
    .= cD(f1,h)/.x - cD(f2,h)/.x by Th5
    .= cD(cdif(f1,h).0,h)/.x - cD(f2,h)/.x by Def8
    .= cD(cdif(f1,h).0,h)/.x - cD(cdif(f2,h).0,h)/.x by Def8
    .= cdif(f1,h).(0+1)/.x - cD(cdif(f2,h).0,h)/.x by Def8
    .= cdif(f1,h).(0+1)/.x - cdif(f2,h).(0+1)/.x by Def8;
    hence thesis;
  end;
A4: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A5: for x holds
    cdif(f1-f2,h).(k+1)/.x = cdif(f1,h).(k+1)/.x - cdif(f2,h).(k+1)/.x;
    let x;
A6: cdif(f1-f2,h).(k+1)/.(x-(2*1.F)"*h)
    = cdif(f1,h).(k+1)/.(x-(2*1.F)"*h) - cdif(f2,h).(k+1)/.(x-(2*1.F)"*h) &
    cdif(f1-f2,h).(k+1)/.(x+(2*1.F)"*h)
    = cdif(f1,h).(k+1)/.(x+(2*1.F)"*h) - cdif(f2,h).(k+1)/.(x+(2*1.F)"*h)
    by A5;
A7: cdif(f1-f2,h).(k+1) is Function of (the carrier of V),(the carrier of W)
    by Th19;
A8: cdif(f2,h).(k+1) is Function of (the carrier of V),(the carrier of W)
    by Th19;
A9: cdif(f1,h).(k+1) is Function of (the carrier of V),(the carrier of W)
    by Th19;
    cdif(f1-f2,h).(k+1+1)/.x = cD(cdif(f1-f2,h).(k+1),h)/.x by Def8
    .= cdif(f1-f2,h).(k+1)/.(x+(2*1.F)"*h)
      - cdif(f1-f2,h).(k+1)/.(x-(2*1.F)"*h) by A7,Th5
    .= ((cdif(f1,h).(k+1)/.(x+(2*1.F)"*h) - cdif(f2,h).(k+1)/.(x+(2*1.F)"*h))
      - cdif(f1,h).(k+1)/.(x-(2*1.F)"*h)) + cdif(f2,h).(k+1)/.(x-(2*1.F)"*h)
    by RLVECT_1:29,A6
    .= (cdif(f1,h).(k+1)/.(x+(2*1.F)"*h) - (cdif(f1,h).(k+1)/.(x-(2*1.F)"*h)
      + cdif(f2,h).(k+1)/.(x+(2*1.F)"*h))) + cdif(f2,h).(k+1)/.(x-(2*1.F)"*h)
    by RLVECT_1:27
    .= cdif(f1,h).(k+1)/.(x+(2*1.F)"*h) - ((cdif(f1,h).(k+1)/.(x-(2*1.F)"*h)
      + cdif(f2,h).(k+1)/.(x+(2*1.F)"*h)) - cdif(f2,h).(k+1)/.(x-(2*1.F)"*h))
    by RLVECT_1:29
    .= cdif(f1,h).(k+1)/.(x+(2*1.F)"*h) - (cdif(f1,h).(k+1)/.(x-(2*1.F)"*h)
      + (cdif(f2,h).(k+1)/.(x+(2*1.F)"*h) - cdif(f2,h).(k+1)/.(x-(2*1.F)"*h)))
    by RLVECT_1:28
    .= (cdif(f1,h).(k+1)/.(x+(2*1.F)"*h) - cdif(f1,h).(k+1)/.(x-(2*1.F)"*h))
      - (cdif(f2,h).(k+1)/.(x+(2*1.F)"*h) - cdif(f2,h).(k+1)/.(x-(2*1.F)"*h))
    by RLVECT_1:27
    .= cD(cdif(f1,h).(k+1),h)/.x - (cdif(f2,h).(k+1)/.(x+(2*1.F)"*h)
      - cdif(f2,h).(k+1)/.(x-(2*1.F)"*h)) by A9,Th5
    .= cD(cdif(f1,h).(k+1),h)/.x - cD(cdif(f2,h).(k+1),h)/.x by A8,Th5
    .= cdif(f1,h).(k+1+1)/.x - cD(cdif(f2,h).(k+1),h)/.x by Def8
    .= cdif(f1,h).(k+1+1)/.x - cdif(f2,h).(k+1+1)/.x by Def8;
    hence thesis;
  end;
  for n holds X[n] from NAT_1:sch 2(A1,A4);
  hence thesis;
end;
