let I be set ; :: thesis: for M being ManySortedSet of I
for SF being MSSubsetFamily of M st EmptyMS I in SF holds
meet SF = EmptyMS I
let M be ManySortedSet of I; :: thesis: for SF being MSSubsetFamily of M st EmptyMS I in SF holds
meet SF = EmptyMS I
let SF be MSSubsetFamily of M; :: thesis: ( EmptyMS I in SF implies meet SF = EmptyMS I )
assume A1: EmptyMS I in SF ; :: thesis: meet SF = EmptyMS I

now :: thesis: for i being object st i in I holds
(meet SF) . i = (EmptyMS I) . i

let i be object ; :: thesis: ( i in I implies (meet SF) . i = (EmptyMS I) . i )
assume A2: i in I ; :: thesis: (meet SF) . i = (EmptyMS I) . i
then consider Q being Subset-Family of (M . i) such that
A3: Q = SF . i and
A4: (meet SF) . i = Intersect Q by Def1;
(EmptyMS I) . i in Q by A1, A2, A3;
then A5: {} in Q ;
(EmptyMS I) . i in SF . i by A1, A2;
then Intersect Q = meet Q by A3, SETFAM_1:def 9;
then Intersect Q = {} by A5, SETFAM_1:4;
hence (meet SF) . i = (EmptyMS I) . i by A4; :: thesis: verum

end;

hence meet SF = EmptyMS I ; :: thesis: verum