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 Th15:
  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: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A2: for x holds
    bdif(f1+f2,h).(k+1)/.x = bdif(f1,h).(k+1)/.x + bdif(f2,h).(k+1)/.x;
    let x;
A3: 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 A2;
A4: bdif(f1+f2,h).(k+1) is Function of V,W by Th12;
A5: bdif(f2,h).(k+1) is Function of V, W by Th12;
A6: 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 A4,Th4
    .= (bdif(f2,h).(k+1)/.x + bdif(f1,h).(k+1)/.x - bdif(f1,h).(k+1)/.(x-h))
      - bdif(f2,h).(k+1)/.(x-h) by RLVECT_1:27,A3
    .= (bdif(f2,h).(k+1)/.x + (bdif(f1,h).(k+1)/.x - bdif(f1,h).(k+1)/.(x-h)))
      - bdif(f2,h).(k+1)/.(x-h) by RLVECT_1:28
    .= (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_1:28
    .= bD(bdif(f1,h).(k+1),h)/.x + (bdif(f2,h).(k+1)/.x
      - bdif(f2,h).(k+1)/.(x-h)) by A6,Th4
    .= bD(bdif(f1,h).(k+1),h)/.x + bD(bdif(f2,h).(k+1),h)/.x by A5,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;
A7: X[0]
  proof
    let x;
    reconsider xx = x, h as Element of V;
B0: dom (f1+f2)= dom f1 /\ dom f2 by VFUNCT_1:def 1
    .= (the carrier of V) /\ dom f2 by FUNCT_2:def 1
    .= (the carrier of V) /\ (the carrier of V) by FUNCT_2:def 1
    .= the carrier of V;
    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/.xx + f2/.xx - (f1+f2)/.(xx-h) by B0, VFUNCT_1:def 1
    .= (f1/.x + f2/.x) - (f1/.(x-h) + f2/.(x-h)) by B0,VFUNCT_1:def 1
    .= ((f1/.x + f2/.x) - f1/.(x-h)) - f2/.(x-h) by RLVECT_1:27
    .= (f2/.x + (f1/.x - f1/.(x-h))) - f2/.(x-h) by RLVECT_1:28
    .= (f1/.x - f1/.(x-h)) + (f2/.x - f2/.(x-h)) by RLVECT_1:28
    .= 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;
  for n holds X[n] from NAT_1:sch 2(A7,A1);
  hence thesis;
end;
