let I be set ; :: thesis: for A, B being ManySortedSet of I st A c= B holds
bool A c= bool B
let A, B be ManySortedSet of I; :: thesis: ( A c= B implies bool A c= bool B )
assume A1: A c= B ; :: thesis: bool A c= bool B
let i be object ; :: according to PBOOLE:def 2 :: thesis: ( not i in I or (bool A) . i c= (bool B) . i )
assume A2: i in I ; :: thesis: (bool A) . i c= (bool B) . i
then A3: A . i c= B . i by A1;
( (bool A) . i = bool (A . i) & (bool B) . i = bool (B . i) ) by A2, Def1;
hence (bool A) . i c= (bool B) . i by A3, ZFMISC_1:67; :: thesis: verum