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,x+h,x+2*h!] = (fdif(f,h).2.x)/(2*h^2)
proof
A1: fdif(f,h).1 is Function of REAL,REAL by DIFF_1:2;
  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,x+h,x+2*h!] = [!f,x+2*h,x+h,x!] by DIFF_1:34
    .= ([!f,x+h,x+2*h!]-[!f,x+h,x!])/((x+2*h)-x) by DIFF_1:29
    .= ([!f,x+h,(x+h)+h!]-[!f,x,x+h!])/((x+2*h)-x) by DIFF_1:29
    .= ((fdif(f,h).1.(x+h))/h-[!f,x,x+h!])/((x+2*h)-x) by Th1
    .= ((fdif(f,h).1.(x+h))/h -(fdif(f,h).1.x)/h)/((x+2*h)-x) by Th1
    .= (fdif(f,h).1.(x+h)-(fdif(f,h).1.x)) /h/((x+2*h)-x) by XCMPLX_1:120
    .= (fD(fdif(f,h).1,h).x/h)/(2*h) by A1,DIFF_1:3
    .= ((fdif(f,h).(1+1).x)/h)/(2*h) by DIFF_1:def 6
    .= (fdif(f,h).2.x)/(h*(2*h)) by XCMPLX_1:78
    .= (fdif(f,h).2.x)/(2*(h^2));
  hence thesis;
end;
