reserve n,m for Element of NAT;
reserve h,k,r,r1,r2,x,x0,x1,x2,x3 for Real;
reserve f,f1,f2 for Function of REAL,REAL;

theorem
  h<>0 implies [!f,x-h,x,x+h!] = (cdif(f,h).2.x)/(2*h*h)
proof
  assume h<>0;
  then x-h<>x & x-h<>x+h & x<>x+h;
  then
A1: x-h, x, x+h are_mutually_distinct by ZFMISC_1:def 5;
A2: cdif(f,h).1 is Function of REAL,REAL by DIFF_1:19;
  [!f,x-h,x,x+h!] = [!f,x+h,x,x-h!] by A1,DIFF_1:34
    .= ([!f,x,x+h!]-[!f,x,x-h!])/((x+h)-(x-h)) by DIFF_1:29
    .= ([!f,(x+h/2)-h/2,(x+h/2)+h/2!]
       -[!f,(x-h/2)-h/2,(x-h/2)+h/2!])/((x+h)-(x-h)) by DIFF_1:29
    .= ((cdif(f,h).1.(x+h/2))/h
       -[!f,(x-h/2)-h/2,(x-h/2)+h/2!])/((x+h)-(x-h)) by Th23
    .= ((cdif(f,h).1.(x+h/2))/h
       -(cdif(f,h).1.(x-h/2))/h)/((x+h)-(x-h)) by Th23
    .= ((cdif(f,h).1.(x+h/2)-cdif(f,h).1.(x-h/2)))/h/((x+h)-(x-h))
    .= ((cD(cdif(f,h).1,h).x)/h)/(2*h) by A2,DIFF_1:5
    .= ((cdif(f,h).(1+1).x)/h)/(2*h) by DIFF_1:def 8
    .= (cdif(f,h).2.x)/(2*h*h) by XCMPLX_1:78;
  hence thesis;
end;
