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

theorem Th34:
  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 Def6;
      hence x in A or x in A1.k by XBOOLE_0:def 3;
    end;
    per cases by A2;
    suppose
      x in A;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose
      x in A1.k;
      then x in Intersection A1 by PROB_1:13;
      hence thesis by XBOOLE_0:def 3;
    end;
  end;
  let x be object;
  assume
A3: x in A \/ Intersection A1;
  per cases by A3,XBOOLE_0:def 3;
  suppose
A4: x in A;
    now
      let k;
      x in (A \/ A1.k) by A4,XBOOLE_0:def 3;
      hence x in (A (\/) A1).k by Def6;
    end;
    hence thesis by PROB_1:13;
  end;
  suppose
A5: x in Intersection A1;
    now
      let k;
      x in A1.k by A5,PROB_1:13;
      then x in A \/ A1.k by XBOOLE_0:def 3;
      hence x in (A (\/) A1).k by Def6;
    end;
    hence thesis by PROB_1:13;
  end;
end;
