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

now :: thesis: for i being object st i in I holds
(X (/\) Y) . i = {}

let i be object ; :: 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;
thus (X (/\) Y) . i = (X . i) /\ (Y . i) by A2, Def5
.= {} by A3 ; :: thesis: verum

end;

hence X (/\) Y = EmptyMS I by Th6; :: thesis: verum