let I be set ; :: thesis: for M, A, B being ManySortedSet of
for SF being MSSubsetFamily of M st A in SF & A c= B holds
meet SF c= B
let M, A, B be ManySortedSet of ; :: thesis: for SF being MSSubsetFamily of M st A in SF & A c= B holds
meet SF c= B
let SF be MSSubsetFamily of M; :: thesis: ( A in SF & A c= B implies meet SF c= B )
assume that
A1: A in SF and
A2: A c= B ; :: thesis: meet SF c= B
let i be set ; :: according to PBOOLE:def 5 :: thesis: ( not i in I or (meet SF) . i c= B . i )
assume A3: i in I ; :: thesis: (meet SF) . i c= B . i
then consider Q being Subset-Family of (M . i) such that
A4: Q = SF . i and
A5: (meet SF) . i = Intersect Q by Def2;
A6: A . i in SF . i by A1, A3, PBOOLE:def 4;
then A7: Intersect Q = meet Q by A4, SETFAM_1:def 10;
A . i c= B . i by A2, A3, PBOOLE:def 5;
hence (meet SF) . i c= B . i by A4, A5, A6, A7, SETFAM_1:8; :: thesis: verum