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
  bdif(f1-f2,h).(n+1)/.x = bdif(f1,h).(n+1)/.x - bdif(f2,h).(n+1)/.x
proof
  defpred X[Nat] means
  for x holds bdif(f1-f2,h).($1+1)/.x = bdif(f1,h).($1+1)/.x
  - bdif(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;
    bdif(f1-f2,h).(0+1)/.x = bD(bdif(f1-f2,h).0,h)/.x by Def7
    .= bD(f1-f2,h)/.x by Def7
    .= (f1-f2)/.x - (f1-f2)/.(x-h) by Th4
    .= f1/.x - f2/.x - (f1-f2)/.(x-h) by A2,VFUNCT_1:def 2
    .= (f1/.x - f2/.x) - (f1/.(x-h) - f2/.(x-h)) by A3,VFUNCT_1:def 2
    .= (f1/.x - f2/.x - f1/.(x-h)) + f2/.(x-h) by RLVECT_1:29
    .= (f1/.x - (f1/.(x-h) + f2/.x)) + f2/.(x-h) by RLVECT_1:27
    .= ((f1/.x - f1/.(x-h)) - f2/.x) + f2/.(x-h) by RLVECT_1:27
    .= (f1/.x - f1/.(x-h)) -(f2/.x - f2/.(x-h)) by RLVECT_1:29
    .= bD(f1,h)/.x - (f2/.x - f2/.(x-h)) by Th4
    .= bD(f1,h)/.x - bD(f2,h)/.x by Th4
    .= bD(bdif(f1,h).0,h)/.x - bD(f2,h)/.x by Def7
    .= bD(bdif(f1,h).0,h)/.x - bD(bdif(f2,h).0,h)/.x by Def7
    .= bdif(f1,h).(0+1)/.x - bD(bdif(f2,h).0,h)/.x by Def7
    .= bdif(f1,h).(0+1)/.x - bdif(f2,h).(0+1)/.x by Def7;
    hence thesis;
  end;
A4: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A5: for x holds
    bdif(f1-f2,h).(k+1)/.x = bdif(f1,h).(k+1)/.x - bdif(f2,h).(k+1)/.x;
    let x;
A6: bdif(f1-f2,h).(k+1)/.x = bdif(f1,h).(k+1)/.x - bdif(f2,h).(k+1)/.x &
    bdif(f1-f2,h).(k+1)/.(x-h)
    = bdif(f1,h).(k+1)/.(x-h) - bdif(f2,h).(k+1)/.(x-h) by A5;
A7: bdif(f1-f2,h).(k+1) is Function of V,W by Th12;
A8: bdif(f2,h).(k+1) is Function of V,W by Th12;
A9: bdif(f1,h).(k+1) is Function of V,W by Th12;
    bdif(f1-f2,h).(k+1+1)/.x = bD(bdif(f1-f2,h).(k+1),h)/.x by Def7
    .= bdif(f1-f2,h).(k+1)/.x - bdif(f1-f2,h).(k+1)/.(x-h) by A7,Th4
    .= (bdif(f1,h).(k+1)/.x - bdif(f2,h).(k+1)/.x - bdif(f1,h).(k+1)/.(x-h))
      + bdif(f2,h).(k+1)/.(x-h) by RLVECT_1:29,A6
    .= (bdif(f1,h).(k+1)/.x - (bdif(f1,h).(k+1)/.(x-h) + bdif(f2,h).(k+1)/.x))
      + bdif(f2,h).(k+1)/.(x-h) by RLVECT_1:27
    .= ((bdif(f1,h).(k+1)/.x - bdif(f1,h).(k+1)/.(x-h)) - bdif(f2,h).(k+1)/.x)
      + bdif(f2,h).(k+1)/.(x-h) by RLVECT_1:27
    .= (bdif(f1,h).(k+1)/.x - bdif(f1,h).(k+1)/.(x-h)) - (bdif(f2,h).(k+1)/.x
      - bdif(f2,h).(k+1)/.(x-h)) by RLVECT_1:29
    .= bD(bdif(f1,h).(k+1),h)/.x - (bdif(f2,h).(k+1)/.x
      - bdif(f2,h).(k+1)/.(x-h)) by A9,Th4
    .= bD(bdif(f1,h).(k+1),h)/.x - bD(bdif(f2,h).(k+1),h)/.x by A8,Th4
    .= bdif(f1,h).(k+1+1)/.x - bD(bdif(f2,h).(k+1),h)/.x by Def7
    .= bdif(f1,h).(k+1+1)/.x - bdif(f2,h).(k+1+1)/.x by Def7;
    hence thesis;
  end;
  for n holds X[n] from NAT_1:sch 2(A1,A4);
  hence thesis;
end;
