let X be set ; :: thesis: for A1, A2 being SetSequence of X holds Intersection (A1 (\) A2) c= (Intersection A1) \ (Intersection A2)
let A1, A2 be SetSequence of X; :: thesis: Intersection (A1 (\) A2) c= (Intersection A1) \ (Intersection A2)
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in Intersection (A1 (\) A2) or x in (Intersection A1) \ (Intersection A2) )
assume A1: x in Intersection (A1 (\) A2) ; :: thesis: x in (Intersection A1) \ (Intersection A2)

A2: now :: thesis: for k being Nat holds
( x in A1 . k & not x in A2 . k )

let k be Nat; :: thesis: ( x in A1 . k & not x in A2 . k )
x in (A1 (\) A2) . k by A1, PROB_1:13;
then x in (A1 . k) \ (A2 . k) by Def3;
hence ( x in A1 . k & not x in A2 . k ) by XBOOLE_0:def 5; :: thesis: verum

end;

then not x in A2 . 0 ;
then A3: not x in Intersection A2 by PROB_1:13;
x in Intersection A1 by A2, PROB_1:13;
hence x in (Intersection A1) \ (Intersection A2) by A3, XBOOLE_0:def 5; :: thesis: verum