let I be set ; :: thesis: for X, Y being ManySortedSet of I st X misses Y holds
X \ Y = X
let X, Y be ManySortedSet of I; :: thesis: ( X misses Y implies X \ Y = X )
assume A1: X misses Y ; :: thesis: X \ Y = X
let i be set ; :: according to PBOOLE:def 10 :: thesis: ( i in I implies (X \ Y) . i = X . i )
assume A2: i in I ; :: thesis: (X \ Y) . i = X . i
then A3: (X \ Y) . i = (X . i) \ (Y . i) by Def6;
X . i misses Y . i by A1, A2, Def8;
hence (X \ Y) . i = X . i by A3, XBOOLE_1:83; :: thesis: verum