 reserve n,m,i,p for Nat,
         h,k,r,r1,r2,x,x0,x1,x2,x3 for Real;
 reserve f,f1,f2,g for Function of REAL,REAL;

theorem
  x+h/2>0 & x-h/2>0 implies cD(ln,h).x = ln.(1+h/(x-h/2))
proof
  set f=ln;
  assume
A1: x+h/2>0 & x-h/2>0;
A2: x+h/2 in right_open_halfline(0)
  proof
    x+h/2 in {g where g is Real: 0<g} by A1;
    hence thesis by XXREAL_1:230;
  end;
A3: x-h/2 in right_open_halfline(0)
  proof
    x-h/2 in {g where g is Real: 0<g} by A1;
    hence thesis by XXREAL_1:230;
  end;
  cD(f,h).x = ln.(x+h/2) - ln.(x-h/2) by A2,A3,DIFF_1:39,TAYLOR_1:18
    .= ln.((x-h/2+h)/(x-h/2)) by A1,Th4
    .= ln.((x-h/2)/(x-h/2)+h/(x-h/2))
    .= ln.(1+h/(x-h/2)) by A1,XCMPLX_1:60;
  hence thesis;
end;
