let I be set ; :: thesis: for M being ManySortedSet of I
for A, B being SubsetFamily of M holds MSUnion (A /\ B) c= (MSUnion A) (/\) (MSUnion B)
let M be ManySortedSet of I; :: thesis: for A, B being SubsetFamily of M holds MSUnion (A /\ B) c= (MSUnion A) (/\) (MSUnion B)
let A, B be SubsetFamily of M; :: thesis: MSUnion (A /\ B) c= (MSUnion A) (/\) (MSUnion B)
reconsider MAB = MSUnion (A /\ B) as ManySortedSet of I ;
reconsider MA = MSUnion A as ManySortedSet of I ;
reconsider MB = MSUnion B as ManySortedSet of I ;
for i being object st i in I holds
MAB . i c= (MA (/\) MB) . i

proof

let i be object ; :: thesis: ( i in I implies MAB . i c= (MA (/\) MB) . i )
assume A1: i in I ; :: thesis: MAB . i c= (MA (/\) MB) . i
then A2: ( MA . i = union { (f . i) where f is Element of Bool M : f in A } & MB . i = union { (f . i) where f is Element of Bool M : f in B } ) by Def2;
A3: MAB . i = union { (f . i) where f is Element of Bool M : f in A /\ B } by A1, Def2;
for v being object st v in MAB . i holds
v in (MA (/\) MB) . i

proof

let v be object ; :: thesis: ( v in MAB . i implies v in (MA (/\) MB) . i )
assume v in MAB . i ; :: thesis: v in (MA (/\) MB) . i
then consider w being set such that
A4: v in w and
A5: w in { (f . i) where f is Element of Bool M : f in A /\ B } by A3, TARSKI:def 4;
consider g being Element of Bool M such that
A6: w = g . i and
A7: g in A /\ B by A5;
g in B by A7, XBOOLE_0:def 4;
then w in { (f . i) where f is Element of Bool M : f in B } by A6;
then A8: v in union { (f . i) where f is Element of Bool M : f in B } by A4, TARSKI:def 4;
g in A by A7, XBOOLE_0:def 4;
then w in { (f . i) where f is Element of Bool M : f in A } by A6;
then v in union { (f . i) where f is Element of Bool M : f in A } by A4, TARSKI:def 4;
then v in (MA . i) /\ (MB . i) by A2, A8, XBOOLE_0:def 4;
hence v in (MA (/\) MB) . i by A1, PBOOLE:def 5; :: thesis: verum

end;

hence MAB . i c= (MA (/\) MB) . i ; :: thesis: verum

end;

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