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

let i be object ; :: 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, Th20
.= (A . i) \/ (B . i) by ZFMISC_1:75
.= (A (\/) B) . i by A2, PBOOLE:def 4 ;
:: thesis: verum

end;

hence union |:SF:| = A (\/) B ; :: thesis: verum