reserve X for non empty TopSpace;
reserve x for Point of X;
reserve U1 for Subset of X;

theorem
  for U1 being Subset of X holds U1 is open iff for x st x in U1 ex A
  being Subset of X st A is a_neighborhood of x & A c= U1
proof
  let U1 be Subset of X;
  now
    assume
A1: for x st x in U1 ex A being Subset of X st A is a_neighborhood of
    x & A c= U1;
    for x being set holds x in U1 iff ex V being Subset of X st V is open
    & V c= U1 & x in V
    proof
      let x be set;
      thus x in U1 implies ex V being Subset of X st V is open & V c= U1 & x
      in V
      proof
        assume
A2:     x in U1;
        then reconsider x as Point of X;
        consider S being Subset of X such that
A3:     S is a_neighborhood of x and
A4:     S c= U1 by A1,A2;
        consider V being Subset of X such that
A5:     V is open & V c= S & x in V by A3,Th6;
        take V;
        thus thesis by A4,A5;
      end;
      given V being Subset of X such that
      V is open and
A6:   V c= U1 & x in V;
      thus thesis by A6;
    end;
    hence U1 is open by TOPS_1:25;
  end;
  hence thesis by Th3;
end;
