let X be set ; :: thesis: for b1, b2 being bag of X holds
( b2 divides b1 iff lcm (b1,b2) = b1 )
let b1, b2 be bag of X; :: thesis: ( b2 divides b1 iff lcm (b1,b2) = b1 )

now

assume A1: b2 divides b1 ; :: thesis: lcm (b1,b2) = b1

now

let k be set ; :: thesis: b1 . k = max ((b1 . k),(b2 . k))
b2 . k <= b1 . k by A1, PRE_POLY:def 11;
hence b1 . k = max ((b1 . k),(b2 . k)) by XXREAL_0:def 10; :: thesis: verum

end;

hence lcm (b1,b2) = b1 by Def2; :: thesis: verum

end;

hence ( b2 divides b1 iff lcm (b1,b2) = b1 ) by Th7; :: thesis: verum