
theorem LMC31H1:
  for x,y be Real st number_e < x & x <= y holds
    x/log(2,x) <= y/log(2,y)
  proof
    let x,y be Real;
    assume AS: number_e < x & x <= y;
    consider f be PartFunc of REAL,REAL such that
    A11: right_open_halfline number_e = dom f and
    A12:(for x be Real st x in dom f holds f.x = x/log(2,x) ) and
    f is_differentiable_on right_open_halfline number_e and
    (for x0 being Real st
    x0 in right_open_halfline number_e holds 0 <= diff (f,x0) ) and
    A15: f is non-decreasing by LMC31H2;
    number_e < y by AS,XXREAL_0:2; then
    x in {g where g is Real : number_e<g} &
    y in {g where g is Real : number_e<g} by AS; then
    A3: x in dom f & y in dom f by A11,XXREAL_1:230; then
    A4: f .x <= f .y by AS,A15,VALUED_0:def 15;
    f.x =x/log(2,x) by A12,A3;
    hence thesis by A4,A12,A3;
  end;
