 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>0 & x-h>0 implies bD(ln,h).x = ln.(1+h/(x-h))
proof
  set f=ln;
  assume
A1: x>0 & x-h>0;
A2: x in right_open_halfline(0)
  proof
    x in {g where g is Real: 0<g} by A1;
    hence thesis by XXREAL_1:230;
  end;
A3: x-h in right_open_halfline(0)
  proof
    x-h in {g where g is Real: 0<g} by A1;
    hence thesis by XXREAL_1:230;
  end;
  bD(f,h).x = ln.x - ln.(x-h) by A2,A3,DIFF_1:38,TAYLOR_1:18
    .= ln.(x/(x-h)) by A1,Th4
    .= ln.((x-h)/(x-h)+h/(x-h))
    .= ln.(1+h/(x-h)) by A1,XCMPLX_1:60;
  hence thesis;
end;
