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)
let i be object ; :: according to PBOOLE:def 10 :: 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 Def6
.= (X . i) \ ((Y . i) \ (Z . i)) by A1, Def6
.= ((X . i) \ (Y . i)) \/ ((X . i) /\ (Z . i)) by XBOOLE_1:52
.= ((X . i) \ (Y . i)) \/ ((X (/\) Z) . i) by A1, Def5
.= ((X (\) Y) . i) \/ ((X (/\) Z) . i) by A1, Def6
.= ((X (\) Y) (\/) (X (/\) Z)) . i by A1, Def4 ;
:: thesis: verum