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 :: thesis: for i being object st i in I holds
(union (A (\/) B)) . i = ((union A) (\/) (union B)) . i

let i be object ; :: 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_1:78
.= ((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 4 ;
:: thesis: verum

end;

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