let I be set ; :: thesis: for A, B being ManySortedSet of holds (bool A) \/ (bool B) c= bool (A \/ B)
let A, B be ManySortedSet of ; :: thesis: (bool A) \/ (bool B) c= bool (A \/ B)
let i be set ; :: according to PBOOLE:def 5 :: thesis: ( not i in I or ((bool A) \/ (bool B)) . i c= (bool (A \/ B)) . i )
assume A1: i in I ; :: thesis: ((bool A) \/ (bool B)) . i c= (bool (A \/ B)) . i
then A2: ((bool A) \/ (bool B)) . i = ((bool A) . i) \/ ((bool B) . i) by PBOOLE:def 7
.= (bool (A . i)) \/ ((bool B) . i) by A1, Def1
.= (bool (A . i)) \/ (bool (B . i)) by A1, Def1 ;
(bool (A \/ B)) . i = bool ((A . i) \/ (B . i)) by A1, Lm2;
hence ((bool A) \/ (bool B)) . i c= (bool (A \/ B)) . i by A2, ZFMISC_1:81; :: thesis: verum