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 /\ Y) . i) /\ (Z . i) by Def8
.= ((X . i) /\ (Y . i)) /\ (Z . i) by A1, Def8
.= (X . i) /\ ((Y . i) /\ (Z . i)) by XBOOLE_1:16
.= (X . i) /\ ((Y /\ Z) . i) by A1, Def8
.= (X /\ (Y /\ Z)) . i by A1, Def8 ;
:: thesis: verum

end;

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