reserve n,m,k for Nat,
  x,X for set,
  A for Subset of X,
  A1,A2 for SetSequence of X;

theorem Th14:
  Intersection (A1 (\) A2) c= Intersection A1 \ Intersection A2
proof
  let x be object;
  assume
A1: x in Intersection (A1 (\) A2);
A2: now
    let 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;
  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 thesis by A3,XBOOLE_0:def 5;
end;
