( ( r <= 0 implies I = [.0,+infty.[ ) & ( r > 0 implies I = [.0,r.] ) ) by Def555;
then reconsider I = I as Element of Borel_Sets by FINANCE1:3, FINANCE1:8;
( A is Event of Borel_Sets & I is Event of Borel_Sets ) ;
hence A /\ I is Element of Borel_Sets by PROB_1:19; :: thesis: verum