let I be set ; :: thesis: for M, A, B being ManySortedSet of I
for SF being MSSubsetFamily of M st A in SF & A /\ B = [[0]] I holds
(meet SF) /\ B = [[0]] I
let M, A, B be ManySortedSet of I; :: thesis: for SF being MSSubsetFamily of M st A in SF & A /\ B = [[0]] I holds
(meet SF) /\ B = [[0]] I
let SF be MSSubsetFamily of M; :: thesis: ( A in SF & A /\ B = [[0]] I implies (meet SF) /\ B = [[0]] I )
assume that
A1: A in SF and
A2: A /\ B = [[0]] I ; :: thesis: (meet SF) /\ B = [[0]] I

now

let i be set ; :: thesis: ( i in I implies ((meet SF) /\ B) . i = ([[0]] I) . i )
assume A3: i in I ; :: thesis: ((meet SF) /\ B) . i = ([[0]] I) . 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 1;
(A . i) /\ (B . i) = ([[0]] I) . i by A2, A3, PBOOLE:def 5;
then (A . i) /\ (B . i) = {} by PBOOLE:5;
then A . i misses B . i by XBOOLE_0:def 7;
then meet Q misses B . i by A4, A6, SETFAM_1:8;
then (meet Q) /\ (B . i) = {} by XBOOLE_0:def 7;
then A7: (meet Q) /\ (B . i) = ([[0]] I) . i by PBOOLE:5;
Intersect Q = meet Q by A4, A6, SETFAM_1:def 9;
hence ((meet SF) /\ B) . i = ([[0]] I) . i by A3, A5, A7, PBOOLE:def 5; :: thesis: verum

end;

hence (meet SF) /\ B = [[0]] I by PBOOLE:3; :: thesis: verum