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(f,h).n)/.x = (bdif(f,h).n)/.(x+n*h)
proof
  defpred X[Nat] means
  for x holds (fdif(f,h).$1)/.x = (bdif(f,h).$1)/.(x+$1*h);
A1: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A2: for x holds (fdif(f,h).k)/.x = (bdif(f,h).k)/.(x+k*h);
    let x;
A3: (fdif(f,h).k)/.(x+h) = (bdif(f,h).k)/.(x+h+k*h) by A2;
    reconsider fdk = fdif(f,h).k as Function of (the carrier of V),
    (the carrier of W) by Th2;
N2: k*h + h = k*h + 1*h by BINOM:13
    .= (k+1)*h by BINOM:15;
N3: k*h = k*h + 0.V by RLVECT_1:4
    .=k*h + (h-h) by RLVECT_1:15
    .= (k+1)*h - h by N2,RLVECT_1:28;
A5: bdif(f,h).k is Function of (the carrier of V),(the carrier of W)
    by Th12;
    (fdif(f,h).(k+1))/.x = fD(fdk,h)/.x by Def6
    .= (fdk)/.(x+h) - (fdk)/.x by Th3
    .= (bdif(f,h).k)/.(x+h+k*h) - (bdif(f,h).k)/.(x+k*h) by A2,A3
    .= (bdif(f,h).k)/.(x+(h+k*h)) - (bdif(f,h).k)/.(x+k*h) by RLVECT_1:def 3
    .= (bdif(f,h).k)/.(x+(k+1)*h) - (bdif(f,h).k)/.((x+(k+1)*h)-h)
    by RLVECT_1:28,N3,N2
    .= bD(bdif(f,h).k,h)/.(x+(k+1)*h) by A5,Th4
    .= (bdif(f,h).(k+1))/.(x+(k+1)*h) by Def7;
    hence thesis;
  end;
A6: X[0]
  proof
    let x;
    (fdif(f,h).0)/.x = f/.x by Def6
    .= (bdif(f,h).0)/.x by Def7
    .= (bdif(f,h).0)/.(x+0.V) by RLVECT_1:4
    .= (bdif(f,h).0)/.(x+0*h) by BINOM:12;
    hence thesis;
  end;
  for n holds X[n] from NAT_1:sch 2(A6,A1);
  hence thesis;
end;
