reserve h,r,r1,r2,x0,x1,x2,x3,x4,x5,x,a,b,c,k for Real,
  f,f1,f2 for Function of REAL,REAL;

theorem
  h<>0 implies [!f,x-2*h,x-h,x!] = (bdif(f,h).2.x)/(2*h^2)
proof
  set y=x-h;
A1: bdif(f,h).1 is Function of REAL,REAL by DIFF_1:12;
  assume
A2: h<>0;
  then
A3: x-h<>x-2*h;
  x<>x-h & x<>x-2*h by A2;
  then x,x-h,x-2*h are_mutually_distinct by A3,ZFMISC_1:def 5;
  then [!f,x-2*h,x-h,x!] = [!f,x,x-h,x-2*h!] by DIFF_1:34
    .= ([!f,x-h,x!]-[!f,x-h,x-2*h!])/(x-(x-2*h)) by DIFF_1:29
    .= ((bdif(f,h).1.x)/h-[!f,x-h,x-2*h!])/(x-(x-2*h)) by Th3
    .= ((bdif(f,h).1.x)/h-[!f,y-h,y!])/(x-(x-2*h)) by DIFF_1:29
    .= ((bdif(f,h).1.x)/h-(bdif(f,h).1.(x-h))/h)/(x-(x-2*h)) by Th3
    .= ((bdif(f,h).1.x-bdif(f,h).1.(x-h))/h) /(x-(x-2*h)) by XCMPLX_1:120
    .= (bD(bdif(f,h).1,h).x/h)/(2*h) by A1,DIFF_1:4
    .= ((bdif(f,h).(1+1).x)/h)/(2*h) by DIFF_1:def 7
    .= (bdif(f,h).2.x)/(h*(2*h)) by XCMPLX_1:78
    .= (bdif(f,h).2.x)/(2*(h^2));
  hence thesis;
end;
