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

theorem Th12:
  Intersection (A1 (/\) A2) = Intersection A1 /\ Intersection A2
proof
  thus Intersection (A1 (/\) A2) c= Intersection A1 /\ Intersection A2
  proof
    let x be object;
    assume
A1: x in Intersection (A1 (/\) A2);
    now
      let k;
      x in (A1 (/\) A2).k by A1,PROB_1:13;
      then x in (A1.k /\ A2.k) by Def1;
      hence x in A1.k & x in A2.k by XBOOLE_0:def 4;
    end;
    then x in Intersection A1 & x in Intersection A2 by PROB_1:13;
    hence thesis by XBOOLE_0:def 4;
  end;
  let x be object;
  assume x in Intersection A1 /\ Intersection A2;
  then
A2: x in Intersection A1 & x in Intersection A2 by XBOOLE_0:def 4;
  now
    let k;
    x in A1.k & x in A2.k by A2,PROB_1:13;
    then x in A1.k /\ A2.k by XBOOLE_0:def 4;
    hence x in (A1 (/\) A2).k by Def1;
  end;
  hence thesis by PROB_1:13;
end;
