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

now

let i be set ; :: thesis: ( i in I implies (X /\ (Y \ Z)) . i = ((X /\ Y) \ Z) . i )
assume A1: i in I ; :: thesis: (X /\ (Y \ Z)) . i = ((X /\ Y) \ Z) . i
hence (X /\ (Y \ Z)) . i = (X . i) /\ ((Y \ Z) . i) by Def8
.= (X . i) /\ ((Y . i) \ (Z . i)) by A1, Def9
.= ((X . i) /\ (Y . i)) \ (Z . i) by XBOOLE_1:49
.= ((X /\ Y) . i) \ (Z . i) by A1, Def8
.= ((X /\ Y) \ Z) . i by A1, Def9 ;
:: thesis: verum

end;

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