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 Def7
.= ((X . i) \/ (Y . i)) \/ (Z . i) by A1, Def7
.= (X . i) \/ ((Y . i) \/ (Z . i)) by XBOOLE_1:4
.= (X . i) \/ ((Y \/ Z) . i) by A1, Def7
.= (X \/ (Y \/ Z)) . i by A1, Def7 ;
:: thesis: verum

end;

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