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
  1.F <> -1.F implies
  (fdif(f,h).(2*n+1))/.x = (cdif(f,h).(2*n+1))/.(x+n*h+(2*1.F)"*h)
proof
  assume
AS: 1.F <> -1.F;
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;
A52: cdif(f,h).(2*n) is Function of V,W by Th19;
A35: x+n*h + (2*1.F)"*h - (2*1.F)"*h
  = x+n*h + ((2*1.F)"*h-(2*1.F)"*h) by RLVECT_1:28
  .=x+n*h + 0.V by RLVECT_1:15
  .= x+n*h by RLVECT_1:4;
A36: x+n*h + (2*1.F)"*h + (2*1.F)"*h
  = x+n*h + ((2*1.F)"*h+(2*1.F)"*h) by RLVECT_1:def 3
  .= x + (n*h+h) by RLVECT_1:def 3,A34
  .= x + (n*h+1*h) by BINOM:13
  .= x + (n+1)*h by BINOM:15;
A51: fdif(f,h).(2*n) is Function of V,W by Th2;
A11: cdif(f,h).(2*n+1)/.(x+n*h+(2*1.F)"*h)
  = cD(cdif(f,h).(2*n),h)/.(x+n*h+(2*1.F)"*h) by Def8
  .= cdif(f,h).(2*n)/.(x+(n+1)*h)
    - cdif(f,h).(2*n)/.(x+n*h+(2*1.F)"*h-(2*1.F)"*h) by A36,Th5,A52
  .= cdif(f,h).(2*n)/.(x+(n*h+1*h)) - cdif(f,h).(2*n)/.(x+n*h) by BINOM:15,A35
  .= cdif(f,h).(2*n)/.(x+(n*h+h)) - cdif(f,h).(2*n)/.(x+n*h) by BINOM:13
  .= cdif(f,h).(2*n)/.(x+h+n*h) - cdif(f,h).(2*n)/.(x+n*h) by RLVECT_1:def 3
  .= fdif(f,h).(2*n)/.(x+h) - cdif(f,h).(2*n)/.(x+n*h) by LAST0,AS
  .= fdif(f,h).(2*n)/.(x+h) - fdif(f,h).(2*n)/.x by LAST0,AS;
  fdif(f,h).(2*n+1)/.x = fD(fdif(f,h).(2*n),h)/.x by Def6
  .= (fdif(f,h).(2*n))/.(x+h) - (fdif(f,h).(2*n))/.x by Th3,A51;
  hence thesis by A11;
end;
