let I be set ; :: thesis: for X, Y, Z being ManySortedSet of I holds X \ (Y \ Z) = (X \ Y) \/ (X /\ Z)
let X, Y, Z be ManySortedSet of I; :: thesis: X \ (Y \ Z) = (X \ Y) \/ (X /\ Z)

now

let i be set ; :: thesis: ( i in I implies (X \ (Y \ Z)) . i = ((X \ Y) \/ (X /\ Z)) . i )
assume A1: i in I ; :: thesis: (X \ (Y \ Z)) . i = ((X \ Y) \/ (X /\ Z)) . i
hence (X \ (Y \ Z)) . i = (X . i) \ ((Y \ Z) . i) by Def9
.= (X . i) \ ((Y . i) \ (Z . i)) by A1, Def9
.= ((X . i) \ (Y . i)) \/ ((X . i) /\ (Z . i)) by XBOOLE_1:52
.= ((X . i) \ (Y . i)) \/ ((X /\ Z) . i) by A1, Def8
.= ((X \ Y) . i) \/ ((X /\ Z) . i) by A1, Def9
.= ((X \ Y) \/ (X /\ Z)) . i by A1, Def7 ;
:: thesis: verum

end;

hence X \ (Y \ Z) = (X \ Y) \/ (X /\ Z) by Th3; :: thesis: verum