let I be set ; :: thesis: for M, A, B being ManySortedSet of I
for SF being SubsetFamily of M st SF = {A,B} holds
union |:SF:| = A \/ B
let M, A, B be ManySortedSet of I; :: thesis: for SF being SubsetFamily of M st SF = {A,B} holds
union |:SF:| = A \/ B
let SF be SubsetFamily of M; :: thesis: ( SF = {A,B} implies union |:SF:| = A \/ B )
assume A1: SF = {A,B} ; :: thesis: union |:SF:| = A \/ B

now

let i be set ; :: thesis: ( i in I implies (union |:SF:|) . i = (A \/ B) . i )
assume A2: i in I ; :: thesis: (union |:SF:|) . i = (A \/ B) . i
hence (union |:SF:|) . i = union (|:SF:| . i) by MBOOLEAN:def 2
.= union {(A . i),(B . i)} by A1, A2, Th21
.= (A . i) \/ (B . i) by ZFMISC_1:75
.= (A \/ B) . i by A2, PBOOLE:def 4 ;
:: thesis: verum

end;

hence union |:SF:| = A \/ B by PBOOLE:3; :: thesis: verum