reserve Omega for set;
reserve X, Y, Z, p,x,y,z for set;
reserve D, E for Subset of Omega;
reserve f for Function;
reserve m,n for Nat;
reserve r,r1 for Real;
reserve seq for Real_Sequence;
reserve F for Field_Subset of X;
reserve ASeq,BSeq for SetSequence of Omega;
reserve A1 for SetSequence of X;

theorem Th13:
  for x being object holds x in Intersection A1 iff for n holds x in A1.n
proof let x be object;
A1: for n holds X \ (Complement A1).n = A1.n
  proof
    let n;
    X \ (Complement A1).n = ((A1.n)`)` by Def2
      .= A1.n;
    hence thesis;
  end;
A2: (for n holds (x in X & not x in (Complement A1).n)) iff for n holds x in
  A1.n
  proof
    thus (for n holds (x in X & not x in (Complement A1).n)) implies for n
    holds x in A1.n
    proof
      assume
A3:   for n holds x in X & not x in (Complement A1).n;
      let n;
      not x in (Complement A1).n by A3;
      then x in X \ (Complement A1).n by A3,XBOOLE_0:def 5;
      hence thesis by A1;
    end;
    assume
A4: for n holds x in A1.n;
    let n;
    x in A1.n by A4;
    then x in X \ (Complement A1).n by A1;
    hence thesis by XBOOLE_0:def 5;
  end;
  x in X & not x in Union Complement A1 iff x in X & for n holds not x in
  (Complement A1).n by Th12;
  hence thesis by A2,XBOOLE_0:def 5;
end;
