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

theorem Th13:
  Intersection A1 \/ Intersection A2 c= Intersection (A1 (\/) A2)
proof
  let x be object;
  assume
A1: x in Intersection A1 \/ Intersection A2;
  per cases by A1,XBOOLE_0:def 3;
  suppose
A2: x in Intersection A1;
    now
      let k;
      x in A1.k by A2,PROB_1:13;
      then x in (A1.k \/ A2.k) by XBOOLE_0:def 3;
      hence x in (A1 (\/) A2).k by Def2;
    end;
    hence thesis by PROB_1:13;
  end;
  suppose
A3: x in Intersection A2;
    now
      let k;
      x in A2.k by A3,PROB_1:13;
      then x in A1.k \/ A2.k by XBOOLE_0:def 3;
      hence x in (A1 (\/) A2).k by Def2;
    end;
    hence thesis by PROB_1:13;
  end;
end;
