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

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 PBOOLE:def 21
.= [:((X . i) \/ (Y . i)),(Z . i):] by A1, PBOOLE:def 7
.= [:(X . i),(Z . i):] \/ [:(Y . i),(Z . i):] by ZFMISC_1:120
.= ([|X,Z|] . i) \/ [:(Y . i),(Z . i):] by A1, PBOOLE:def 21
.= ([|X,Z|] . i) \/ ([|Y,Z|] . i) by A1, PBOOLE:def 21
.= ([|X,Z|] \/ [|Y,Z|]) . i by A1, PBOOLE:def 7 ;
:: thesis: verum

end;

hence [|(X \/ Y),Z|] = [|X,Z|] \/ [|Y,Z|] by PBOOLE:3; :: thesis: [|Z,(X \/ Y)|] = [|Z,X|] \/ [|Z,Y|]

now

let i be set ; :: thesis: ( i in I implies [|Z,(X \/ Y)|] . i = ([|Z,X|] \/ [|Z,Y|]) . i )
assume A2: i in I ; :: thesis: [|Z,(X \/ Y)|] . i = ([|Z,X|] \/ [|Z,Y|]) . i
hence [|Z,(X \/ Y)|] . i = [:(Z . i),((X \/ Y) . i):] by PBOOLE:def 21
.= [:(Z . i),((X . i) \/ (Y . i)):] by A2, PBOOLE:def 7
.= [:(Z . i),(X . i):] \/ [:(Z . i),(Y . i):] by ZFMISC_1:120
.= ([|Z,X|] . i) \/ [:(Z . i),(Y . i):] by A2, PBOOLE:def 21
.= ([|Z,X|] . i) \/ ([|Z,Y|] . i) by A2, PBOOLE:def 21
.= ([|Z,X|] \/ [|Z,Y|]) . i by A2, PBOOLE:def 7 ;
:: thesis: verum

end;

hence [|Z,(X \/ Y)|] = [|Z,X|] \/ [|Z,Y|] by PBOOLE:3; :: thesis: verum