
theorem Lm6:
  ex g be PartFunc of REAL,REAL st
  dom(g)=right_open_halfline(0) &
  (for x be Real st x in right_open_halfline(0) holds g.x=log(2,x))
  &
  g is_differentiable_on right_open_halfline(0) &
  for x be Real st x in right_open_halfline(0)
  holds g is_differentiable_in x
  & diff(g,x)=(log(2,number_e))/x
  & 0 < diff(g,x)
  proof
    set g = (log(2,number_e)) (#)ln;
    take g;
    thus
    A3: dom g = dom ln by VALUED_1:def 5
    .= right_open_halfline(0) by TAYLOR_1:def 2;
    E1: number_e > 1 by XXREAL_0:2,TAYLOR_1:11;
    thus for d be Real st d in right_open_halfline(0) holds g.d = log(2,d)
    proof
      let d be Real;
      assume A51:d in right_open_halfline(0); then
      reconsider d0 = d as Element of right_open_halfline(0);
      d in {y where y is Real: 0<y} by XXREAL_1:230,A51; then
      E3: ex y be Real st d=y & 0 < y;
      thus g.d = (log(2,number_e)) *ln.d by A3,A51,VALUED_1:def 5
      .= (log(2,number_e))*log(number_e,d0) by TAYLOR_1:def 2
      .= log(2,d) by E1,E3,POWER:56;
    end;
    thus g is_differentiable_on right_open_halfline(0)
      by FDIFF_1:20,A3,TAYLOR_1:18;
    thus for x be Real
    st x in right_open_halfline(0) holds g is_differentiable_in x
    & diff(g,x)=(log(2,number_e))/x
    & 0 < diff(g,x)
    proof
      let x be Real;
      assume A1:x in right_open_halfline(0);
      A2:ln is_differentiable_in x by A1,TAYLOR_1:18;
      hence g is_differentiable_in x by FDIFF_1:15;
      A3: diff(ln,x)=1/x by A1,TAYLOR_1:18;
      A4: diff(g,x) = (log(2,number_e)) * diff(ln,x) by FDIFF_1:15,A2;
      thus diff(g,x) = (1*log(2,number_e))/x by XCMPLX_1:74,A4,A3
      .= log(2,number_e)/x;
      A5: 0 < diff(ln,x) by A1,TAYLOR_1:18;
      log(2,2) < log(2,number_e) by POWER:57,TAYLOR_1:11;
      then 0 < log(2,number_e) by POWER:52;
      hence 0 < diff(g,x) by A4,A5;
    end;
  end;
