
theorem Th4:
  for X being set, b1,b2,b3 be bag of X holds b1 divides b3 & b2
  divides b3 implies lcm(b1,b2) divides b3
proof
  let X being set, b1,b2,b3 be bag of X;
  assume that
A1: b1 divides b3 and
A2: b2 divides b3;
  now
    let k be object;
    assume k in X;
    now
      per cases by XXREAL_0:16;
      case
        max(b1.k,b2.k) = b1.k;
        hence max(b1.k,b2.k) <= b3.k by A1,PRE_POLY:def 11;
      end;
      case
        max(b1.k,b2.k) = b2.k;
        hence max(b1.k,b2.k) <= b3.k by A2,PRE_POLY:def 11;
      end;
    end;
    hence lcm(b1,b2).k <= b3.k by Def2;
  end;
  hence thesis by PRE_POLY:46;
end;
