let X be set ; :: thesis: for A being Subset of X
for A1 being SetSequence of X holds Intersection (A (\/) A1) = A \/ (Intersection A1)
let A be Subset of X; :: thesis: for A1 being SetSequence of X holds Intersection (A (\/) A1) = A \/ (Intersection A1)
let A1 be SetSequence of X; :: thesis: Intersection (A (\/) A1) = A \/ (Intersection A1)
thus Intersection (A (\/) A1) c= A \/ (Intersection A1) :: according to XBOOLE_0:def 10 :: thesis: A \/ (Intersection A1) c= Intersection (A (\/) A1)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in Intersection (A (\/) A1) or x in A \/ (Intersection A1) )
assume A1: x in Intersection (A (\/) A1) ; :: thesis: x in A \/ (Intersection A1)

A2: now :: thesis: for k being Nat holds
( x in A or x in A1 . k )

let k be Nat; :: thesis: ( x in A or x in A1 . k )
x in (A (\/) A1) . k by A1, PROB_1:13;
then x in A \/ (A1 . k) by Def6;
hence ( x in A or x in A1 . k ) by XBOOLE_0:def 3; :: thesis: verum

end;

per cases ( x in A or for k being Nat holds x in A1 . k ) by A2;

suppose x in A ; :: thesis: x in A \/ (Intersection A1)

hence x in A \/ (Intersection A1) by XBOOLE_0:def 3; :: thesis: verum

end;

suppose for k being Nat holds x in A1 . k ; :: thesis: x in A \/ (Intersection A1)

then x in Intersection A1 by PROB_1:13;
hence x in A \/ (Intersection A1) by XBOOLE_0:def 3; :: thesis: verum

end;

end;

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in A \/ (Intersection A1) or x in Intersection (A (\/) A1) )
assume A3: x in A \/ (Intersection A1) ; :: thesis: x in Intersection (A (\/) A1)

per cases ( x in A or x in Intersection A1 ) by A3, XBOOLE_0:def 3;

suppose A4: x in A ; :: thesis: x in Intersection (A (\/) A1)

now :: thesis: for k being Nat holds x in (A (\/) A1) . k

let k be Nat; :: thesis: x in (A (\/) A1) . k
x in A \/ (A1 . k) by A4, XBOOLE_0:def 3;
hence x in (A (\/) A1) . k by Def6; :: thesis: verum

end;

hence x in Intersection (A (\/) A1) by PROB_1:13; :: thesis: verum

end;

suppose A5: x in Intersection A1 ; :: thesis: x in Intersection (A (\/) A1)

now :: thesis: for k being Nat holds x in (A (\/) A1) . k

let k be Nat; :: thesis: x in (A (\/) A1) . k
x in A1 . k by A5, PROB_1:13;
then x in A \/ (A1 . k) by XBOOLE_0:def 3;
hence x in (A (\/) A1) . k by Def6; :: thesis: verum

end;

hence x in Intersection (A (\/) A1) by PROB_1:13; :: thesis: verum

end;

end;