reserve x,y,X for set;

theorem Th12:
  for T being non empty TopSpace for N being net of T for x being
  Point of T holds x in Lim N iff x is_a_convergence_point_of a_filter N, T
proof
  let T be non empty TopSpace;
  let N be net of T;
  set F = a_filter N;
  let x be Point of T;
  thus x in Lim N implies x is_a_convergence_point_of F, T
  proof
    assume
A1: x in Lim N;
    let A be Subset of T;
    assume that
A2: A is open and
A3: x in A;
    A is a_neighborhood of x by A2,A3,CONNSP_2:3;
    then N is_eventually_in A by A1,YELLOW_6:def 15;
    hence thesis;
  end;
  assume
A4: for A being Subset of T st A is open & x in A holds A in F;
  now
    let O be a_neighborhood of x;
    x in Int O by CONNSP_2:def 1;
    then Int O in F by A4;
    then
A5: N is_eventually_in Int O by Th10;
    Int O c= O by TOPS_1:16;
    hence N is_eventually_in O by A5,WAYBEL_0:8;
  end;
  hence thesis by YELLOW_6:def 15;
end;
