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 Def8
.= (X . i) /\ ((Y . i) \/ (Z . i)) by A1, Def7
.= ((X . i) /\ (Y . i)) \/ ((X . i) /\ (Z . i)) by XBOOLE_1:23
.= ((X /\ Y) . i) \/ ((X . i) /\ (Z . i)) by A1, Def8
.= ((X /\ Y) . i) \/ ((X /\ Z) . i) by A1, Def8
.= ((X /\ Y) \/ (X /\ Z)) . i by A1, Def7 ;
:: thesis: verum

end;

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