let I be set ; :: thesis: for B being ManySortedSet of
for A being V2() ManySortedSet of st ( for X being ManySortedSet of st X in A holds
X /\ B = [0] I ) holds
(union A) /\ B = [0] I
let B be ManySortedSet of ; :: thesis: for A being V2() ManySortedSet of st ( for X being ManySortedSet of st X in A holds
X /\ B = [0] I ) holds
(union A) /\ B = [0] I
let A be V2() ManySortedSet of ; :: thesis: ( ( for X being ManySortedSet of st X in A holds
X /\ B = [0] I ) implies (union A) /\ B = [0] I )
assume A1: for X being ManySortedSet of st X in A holds
X /\ B = [0] I ; :: thesis: (union A) /\ B = [0] I

now

let i be set ; :: thesis: ( i in I implies ((union A) /\ B) . i = ([0] I) . i )
assume A2: i in I ; :: thesis: ((union A) /\ B) . i = ([0] I) . i
for X' being set st X' in A . i holds
X' misses B . i

proof

let X' be set ; :: thesis: ( X' in A . i implies X' misses B . i )
assume A3: X' in A . i ; :: thesis: X' misses B . i
consider M being ManySortedSet of such that
A4: M in A by PBOOLE:146;
A5: dom (i .--> X') = {i} by FUNCOP_1:19;
dom (M +* (i .--> X')) = I by A2, Lm1;
then reconsider K = M +* (i .--> X') as ManySortedSet of by PARTFUN1:def 4, RELAT_1:def 18;
i in {i} by TARSKI:def 1;
then A6: K . i = (i .--> X') . i by A5, FUNCT_4:14
.= X' by FUNCOP_1:87 ;
K in A

proof

let j be set ; :: according to PBOOLE:def 4 :: thesis: ( not j in I or K . j in A . j )
assume A7: j in I ; :: thesis: K . j in A . j

now

per cases ( j = i or j <> i ) ;

case j = i ; :: thesis: K . j in A . j

hence K . j in A . j by A3, A6; :: thesis: verum

end;

case j <> i ; :: thesis: K . j in A . j

then not j in dom (i .--> X') by A5, TARSKI:def 1;
then K . j = M . j by FUNCT_4:12;
hence K . j in A . j by A4, A7, PBOOLE:def 4; :: thesis: verum

end;

end;

end;

hence K . j in A . j ; :: thesis: verum

end;

then K /\ B = [0] I by A1;
then (K . i) /\ (B . i) = ([0] I) . i by A2, PBOOLE:def 8;
then X' /\ (B . i) = {} by A6, PBOOLE:5;
hence X' misses B . i by XBOOLE_0:def 7; :: thesis: verum

end;

then union (A . i) misses B . i by ZFMISC_1:98;
then (union (A . i)) /\ (B . i) = {} by XBOOLE_0:def 7;
then ((union A) . i) /\ (B . i) = {} by A2, Def2;
then ((union A) /\ B) . i = {} by A2, PBOOLE:def 8;
hence ((union A) /\ B) . i = ([0] I) . i by PBOOLE:5; :: thesis: verum

end;

hence (union A) /\ B = [0] I by PBOOLE:3; :: thesis: verum