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 Th14:
  bdif(r(#)f,h).(n+1)/.x = r * bdif(f,h).(n+1)/.x
proof
  defpred X[Nat] means
  for x holds bdif(r(#)f,h).($1+1)/.x = r * bdif(f,h).($1+1)/.x;
A1: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A2: for x holds bdif(r(#)f,h).(k+1)/.x = r * bdif(f,h).(k+1)/.x;
    let x;
A3: bdif(r(#)f,h).(k+1)/.x = r * bdif(f,h).(k+1)/.x &
    bdif(r(#)f,h).(k+1)/.(x-h) = r * bdif(f,h).(k+1)/.(x-h) by A2;
A4: bdif(r(#)f,h).(k+1) is Function of V,W by Th12;
A5: bdif(f,h).(k+1) is Function of V,W by Th12;
    bdif(r(#)f,h).(k+1+1)/.x = bD(bdif(r(#)f,h).(k+1),h)/.x by Def7
    .= bdif(r(#)f,h).(k+1)/.x - bdif(r(#)f,h).(k+1)/.(x-h) by A4,Th4
    .= r * (bdif(f,h).(k+1)/.x - bdif(f,h).(k+1)/.(x-h)) by VECTSP_1:23,A3
    .= r * bD(bdif(f,h).(k+1),h)/.x by A5,Th4
    .= r * bdif(f,h).(k+1+1)/.x by Def7;
    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;
    bdif(r(#)f,h).(0+1)/.x = bD(bdif(r(#)f,h).0,h)/.x by Def7
    .= bD(r(#)f,h)/.x by Def7
    .= (r(#)f)/.x - (r(#)f)/.(x-h) by Th4
    .= (r(#)f)/.x - r * f/.(x-h) by A8,Def4X
    .= r * f/.x - r * f/.(x-h) by A7,Def4X
    .= r * (f/.x - f/.(x-h)) by VECTSP_1:23
    .= r * bD(f,h)/.x by Th4
    .= r * bD(bdif(f,h).0,h)/.x by Def7
    .= r * bdif(f,h).(0+1)/.x by Def7;
    hence thesis;
  end;
  for n holds X[n] from NAT_1:sch 2(A6,A1);
  hence thesis;
end;
