let I be set ; :: thesis: for A, B being ManySortedSet of I holds union (A \/ B) = (union A) \/ (union B)
let A, B be ManySortedSet of I; :: thesis: union (A \/ B) = (union A) \/ (union B)

now

let i be set ; :: thesis: ( i in I implies (union (A \/ B)) . i = ((union A) \/ (union B)) . i )
assume A1: i in I ; :: thesis: (union (A \/ B)) . i = ((union A) \/ (union B)) . i
hence (union (A \/ B)) . i = union ((A . i) \/ (B . i)) by Lm6
.= (union (A . i)) \/ (union (B . i)) by ZFMISC_1:96
.= ((union A) . i) \/ (union (B . i)) by A1, Def2
.= ((union A) . i) \/ ((union B) . i) by A1, Def2
.= ((union A) \/ (union B)) . i by A1, PBOOLE:def 7 ;
:: thesis: verum

end;

hence union (A \/ B) = (union A) \/ (union B) by PBOOLE:3; :: thesis: verum