let X be set ; :: thesis: for b1, b2, b3 being bag of X st lcm (b2,b1) divides lcm (b2,b3) holds
lcm (b1,b3) divides lcm (b2,b3)
let b1, b2, b3 be bag of X; :: thesis: ( lcm (b2,b1) divides lcm (b2,b3) implies lcm (b1,b3) divides lcm (b2,b3) )
assume A1: lcm (b2,b1) divides lcm (b2,b3) ; :: thesis: lcm (b1,b3) 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 ; :: thesis: ( k in X implies (lcm (b1,b3)) . k <= (lcm (b2,b3)) . k )
assume k in X ; :: thesis: (lcm (b1,b3)) . k <= (lcm (b2,b3)) . k
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 (lcm (b1,b3)) . k <= (lcm (b2,b3)) . k by Def2; :: thesis: verum

end;

hence lcm (b1,b3) divides lcm (b2,b3) by PRE_POLY:46; :: thesis: verum