let I be set ; :: thesis: for X, Y being ManySortedSet of I st X misses Y holds
X /\ Y = [[0]] I
let X, Y be ManySortedSet of I; :: thesis: ( X misses Y implies X /\ Y = [[0]] I )
assume A1: X misses Y ; :: thesis: X /\ Y = [[0]] I

now

let i be set ; :: thesis: ( i in I implies (X /\ Y) . i = {} )
assume A2: i in I ; :: thesis: (X /\ Y) . i = {}
then A3: X . i misses Y . i by A1, Def11;
thus (X /\ Y) . i = (X . i) /\ (Y . i) by A2, Def8
.= {} by A3, XBOOLE_0:def 7 ; :: thesis: verum

end;

hence X /\ Y = [[0]] I by Th6; :: thesis: verum