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

theorem Th33:
  Intersection (A (/\) A1) = A /\ Intersection A1
proof
  thus 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 Def5;
      hence x in A & x in A1.k by XBOOLE_0:def 4;
    end;
    then x in Intersection A1 by PROB_1:13;
    hence thesis by A2,XBOOLE_0:def 4;
  end;
  let x be object;
  assume
A3: x in A /\ Intersection A1;
  then
A4: x in Intersection A1 by XBOOLE_0:def 4;
  now
    let k;
    x in A & x in A1.k by A3,A4,PROB_1:13,XBOOLE_0:def 4;
    then x in A /\ A1.k by XBOOLE_0:def 4;
    hence x in (A (/\) A1).k by Def5;
  end;
  hence thesis by PROB_1:13;
end;
