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

theorem Th35:
  Intersection (A (\) A1) c= A \ Intersection A1
proof
  let x be object;
  assume
A1: x in Intersection (A (\) A1);
A2: now
    let k;
    x in (A (\) A1).k by A1,PROB_1:13;
    then x in (A \ A1.k) by Def7;
    hence x in A & not x in A1.k by XBOOLE_0:def 5;
  end;
  then not x in A1.0;
  then not x in Intersection A1 by PROB_1:13;
  hence thesis by A2,XBOOLE_0:def 5;
end;
