 reserve n,i,k,m for Nat;
 reserve p for Prime;

theorem LogMono:
  for x,y being Real st 0 < x < y holds
    ln.x < ln.y
  proof
    let x,y be Real;
    assume
A1: 0 < x < y; then
A2: x in right_open_halfline 0 & y in right_open_halfline 0 by XXREAL_1:235;
    number_e > 1 by XXREAL_0:2,TAYLOR_1:11; then
    log (number_e, x) < log (number_e, y) by A1,POWER:57; then
    (log_number_e).x < log (number_e, y) by A2,TAYLOR_1:def 2;
    hence thesis by TAYLOR_1:def 3,TAYLOR_1:def 2,A2;
  end;
