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

now

let i be set ; :: thesis: ( i in I implies ((X \/ Y) \ Z) . i = ((X \ Z) \/ (Y \ Z)) . i )
assume A1: i in I ; :: thesis: ((X \/ Y) \ Z) . i = ((X \ Z) \/ (Y \ Z)) . i
hence ((X \/ Y) \ Z) . i = ((X \/ Y) . i) \ (Z . i) by Def9
.= ((X . i) \/ (Y . i)) \ (Z . i) by A1, Def7
.= ((X . i) \ (Z . i)) \/ ((Y . i) \ (Z . i)) by XBOOLE_1:42
.= ((X . i) \ (Z . i)) \/ ((Y \ Z) . i) by A1, Def9
.= ((X \ Z) . i) \/ ((Y \ Z) . i) by A1, Def9
.= ((X \ Z) \/ (Y \ Z)) . i by A1, Def7 ;
:: thesis: verum

end;

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