let I be set ; :: thesis: for X, Y being ManySortedSet of st X \ Y = X holds
X misses Y
let X, Y be ManySortedSet of ; :: thesis: ( X \ Y = X implies X misses Y )
assume A1: X \ Y = X ; :: thesis: X misses Y
let i be set ; :: according to PBOOLE:def 11 :: thesis: ( i in I implies X . i misses Y . i )
assume i in I ; :: thesis: X . i misses Y . i
then (X . i) \ (Y . i) = X . i by A1, Def9;
hence X . i misses Y . i by XBOOLE_1:83; :: thesis: verum