 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 fD(ln,h).x = ln.(1+h/x)
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;
  fD(f,h).x = ln.(x+h) - ln.x by A2,A3,DIFF_1:1,TAYLOR_1:18
    .= ln.((x+h)/x) by A1,Th4
    .= ln.(x/x+h/x)
    .= ln.(1+h/x) by A1,XCMPLX_1:60;
  hence thesis;
end;
