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

theorem Th6:
  U1 is a_neighborhood of x iff ex V being Subset of X st V is open
  & V c= U1 & x in V
proof
A1: now
    assume U1 is a_neighborhood of x;
    then consider V being non empty Subset of X such that
A2: V is a_neighborhood of x & V is open & V c= U1 by Th5;
    take V;
    thus ex V being Subset of X st V is open & V c= U1 & x in V by A2,Th4;
  end;
  now
    given V being Subset of X such that
A3: V is open & V c= U1 & x in V;
    x in Int U1 by A3,TOPS_1:22;
    hence U1 is a_neighborhood of x by Def1;
  end;
  hence thesis by A1;
end;
