
theorem
  for X being set, b1,b2,b3 being bag of X holds lcm(b2,b1) divides lcm(
  b2,b3) implies lcm(b1,b3) divides lcm(b2,b3)
proof
  let X be set, b1,b2,b3 be bag of X;
  assume
A1: lcm(b2,b1) divides lcm(b2,b3);
  for k being object st k in X holds lcm(b1,b3).k <= lcm(b2,b3).k
  proof
    let k be object;
    assume k in X;
A2: b3.k <= max(b2.k,b3.k) by XXREAL_0:25;
    lcm(b2,b1).k <= lcm(b2,b3).k by A1,PRE_POLY:def 11;
    then max(b2.k,b1.k) <= lcm(b2,b3).k by Def2;
    then
A3: max(b2.k,b1.k) <= max(b2.k,b3.k) by Def2;
    b1.k <= max(b2.k,b1.k) by XXREAL_0:25;
    then b1.k <= max(b2.k,b3.k) by A3,XXREAL_0:2;
    then max(b1.k,b3.k) <= max(b2.k,b3.k) by A2,XXREAL_0:28;
    then max(b1.k,b3.k) <= lcm(b2,b3).k by Def2;
    hence thesis by Def2;
  end;
  hence thesis by PRE_POLY:46;
end;
