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 LAST0:
  1.F <> -1.F implies (fdif(f,h).(2*n))/.x = (cdif(f,h).(2*n))/.(x+n*h)
proof
  assume
AS: 1.F <> -1.F;
  defpred X[Nat] means
  for x holds (fdif(f,h).(2*$1))/.x = (cdif(f,h).(2*$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).(2*k))/.x = cdif(f,h).(2*k)/.(x+k*h);
    let x;
A31: h+h = 1*h+h by BINOM:13
    .= 1*h + 1*h by BINOM:13
    .= (1+1)*h by BINOM:15;
A32: 1.F + 1.F = 1*1.F + 1.F by BINOM:13
    .= 1*1.F + 1*1.F by BINOM:13
    .= (1+1)*1.F by BINOM:15
    .= 2*1.F;
A33: h+h = 1.F*h + h
    .= 1.F*h + 1.F*h
    .= (2*1.F) * h by A32,VECTSP_1:def 15;
A30: 2*1.F <> 0.F
    proof
      assume
A301: 2*1.F = 0.F;
      1.F + 1.F
      = 1*1.F + 1.F by BINOM:13
      .= 1*1.F + 1*1.F by BINOM:13
      .= (1+1)*1.F by BINOM:15
      .= 0.F by A301;
      hence contradiction by AS,RLVECT_1:def 10;
    end;
A34: (2*1.F)"*h + (2*1.F)"*h
    = (2*1.F)"*(h+h) by VECTSP_1:def 14
    .= ((2*1.F)"*(2*1.F))*h by VECTSP_1:def 16,A33
    .= 1.F * h by A30,VECTSP_1:def 10
    .= h;
A35: x+(k+1)*h - (2*1.F)"*h - (2*1.F)"*h
    = x+ (k*h + 1*h) - (2*1.F)"*h - (2*1.F)"*h by BINOM:15
    .= x + k*h + 1*h - (2*1.F)"*h - (2*1.F)"*h by RLVECT_1:def 3
    .= x + k*h + h - (2*1.F)"*h - (2*1.F)"*h by BINOM:13
    .= x + k*h + h - ((2*1.F)"*h + (2*1.F)"*h) by RLVECT_1:27
    .= x + k*h + (h-h) by RLVECT_1:28,A34
    .= x + k*h + 0.V by RLVECT_1:15
    .= x + k*h by RLVECT_1:4;
A36: x+(k+1)*h + (2*1.F)"*h + (2*1.F)"*h
    = x + (k+1)*h + ((2*1.F)"*h + (2*1.F)"*h) by RLVECT_1:def 3
    .= x + ((k+1)*h + h) by RLVECT_1:def 3,A34
    .= x + ((k+1)*h + 1*h) by BINOM:13
    .= x + ((k+1+1)*h) by BINOM:15
    .= x + (k+2)*h;
A3: fdif(f,h).(2*k)/.(x+h+h) = cdif(f,h).(2*k)/.(x+h+h+k*h) by A2
    .= cdif(f,h).(2*k)/.(x+h+(h+k*h)) by RLVECT_1:def 3
    .= cdif(f,h).(2*k)/.(x+(h+(h+k*h))) by RLVECT_1:def 3
    .= cdif(f,h).(2*k)/.(x+(h+h+(k*h))) by RLVECT_1:def 3
    .= cdif(f,h).(2*k)/.(x+(k+2)*h) by BINOM:15,A31;
A4: fdif(f,h).(2*k)/.(x+h) = cdif(f,h).(2*k)/.(x+h+k*h) by A2
    .= cdif(f,h).(2*k)/.(x+1*h+k*h) by BINOM:13
    .= cdif(f,h).(2*k)/.(x+(1*h+k*h)) by RLVECT_1:def 3
    .= cdif(f,h).(2*k)/.(x+(k+1)*h) by BINOM:15;
    set r3 = cdif(f,h).(2*k)/.(x+k*h);
    set r2 = cdif(f,h).(2*k)/.(x+(k+1)*h);
    set r1 = cdif(f,h).(2*k)/.(x+(k+2)*h);
A5: fdif(f,h).(2*k+1) is Function of V,W by Th2;
A6: cdif(f,h).(2*k) is Function of V,W by Th19;
A7: cdif(f,h).(2*k+1) is Function of V,W by Th19;
A8: fdif(f,h).(2*k) is Function of V,W by Th2;
A9: cdif(f,h).(2*k+1)/.(x+(k+1)*h-(2*1.F)"*h)
    = cD(cdif(f,h).(2*k),h)/.(x+(k+1)*h-(2*1.F)"*h) by Def8
    .= cdif(f,h).(2*k)/.(x+(k+1)*h-(2*1.F)"*h+(2*1.F)"*h)
      - cdif(f,h).(2*k)/.(x+(k+1)*h-(2*1.F)"*h-(2*1.F)"*h) by A6,Th5
    .= cdif(f,h).(2*k)/.(x+(k+1)*h-((2*1.F)"*h-(2*1.F)"*h))
      - cdif(f,h).(2*k)/.(x+(k+1)*h-(2*1.F)"*h-(2*1.F)"*h) by RLVECT_1:29
    .= cdif(f,h).(2*k)/.(x+(k+1)*h-0.V)
      - cdif(f,h).(2*k)/.(x+(k+1)*h-(2*1.F)"*h-(2*1.F)"*h) by RLVECT_1:15
    .= cdif(f,h).(2*k)/.(x+(k+1)*h) - cdif(f,h).(2*k)/.(x+k*h)
    by A35,RLVECT_1:13;
A10: cdif(f,h).(2*k+1)/.(x+(k+1)*h+(2*1.F)"*h)
    = cD(cdif(f,h).(2*k),h)/.(x+(k+1)*h+(2*1.F)"*h) by Def8
    .= cdif(f,h).(2*k)/.(x+(k+1)*h+(2*1.F)"*h+(2*1.F)"*h)
      - cdif(f,h).(2*k)/.(x+(k+1)*h+(2*1.F)"*h-(2*1.F)"*h) by A6,Th5
    .= cdif(f,h).(2*k)/.(x+(k+1)*h+(2*1.F)"*h+(2*1.F)"*h)
      - cdif(f,h).(2*k)/.(x+(k+1)*h+((2*1.F)"*h-(2*1.F)"*h)) by RLVECT_1:28
    .= cdif(f,h).(2*k)/.(x+(k+1)*h+(2*1.F)"*h+(2*1.F)"*h)
      - cdif(f,h).(2*k)/.(x+(k+1)*h+0.V) by RLVECT_1:15
    .= cdif(f,h).(2*k)/.(x+(k+2)*h) - cdif(f,h).(2*k)/.(x+(k+1)*h)
    by A36,RLVECT_1:4;
A11: cdif(f,h).(2*(k+1))/.(x+(k+1)*h) = cdif(f,h).(2*k+1+1)/.(x+(k+1)*h)
    .= cD(cdif(f,h).(2*k+1),h)/.(x+(k+1)*h) by Def8
    .= (r1-r2) - (r2-r3) by A7,A10,A9,Th5;
    fdif(f,h).(2*(k+1))/.x = fdif(f,h).(2*k+1+1)/.x
    .= fD(fdif(f,h).(2*k+1),h)/.x by Def6
    .= (fdif(f,h).(2*k+1))/.(x+h) - (fdif(f,h).(2*k+1))/.x by A5,Th3
    .= fD(fdif(f,h).(2*k),h)/.(x+h) - (fdif(f,h).(2*k+1))/.x by Def6
    .= fD(fdif(f,h).(2*k),h)/.(x+h) - fD(fdif(f,h).(2*k),h)/.x by Def6
    .= fdif(f,h).(2*k)/.(x+h+h) - fdif(f,h).(2*k)/.(x+h)
      - fD(fdif(f,h).(2*k),h)/.x by A8,Th3
    .= fdif(f,h).(2*k)/.(x+h+h) - fdif(f,h).(2*k)/.(x+h)
      - (fdif(f,h).(2*k)/.(x+h) - fdif(f,h).(2*k)/.x) by A8,Th3
    .= cdif(f,h).(2*k)/.(x+(k+2)*h) - cdif(f,h).(2*k)/.(x+(k+1)*h)
      - (cdif(f,h).(2*k)/.(x+(k+1)*h) - cdif(f,h).(2*k)/.(x+k*h)) by A2,A3,A4;
    hence thesis by A11;
  end;
A12: X[0]
  proof
    let x;
    (fdif(f,h).(2*0))/.x = f/.x by Def6
    .= (cdif(f,h).(2*0))/.x by Def8
    .= (cdif(f,h).(2*0))/.(x+0.V) by RLVECT_1:4
    .= (cdif(f,h).(2*0))/.(x+0*h) by BINOM:12;
    hence thesis;
  end;
  for n holds X[n] from NAT_1:sch 2(A12,A1);
  hence thesis;
end;
