reserve
  X for non empty set,
  FX for Filter of X,
  SFX for Subset-Family of X;

theorem Th38:
  for T being Hausdorff non empty TopSpace,
      F being Filter of the carrier of T,p,q being Point of T st
  p in lim_filter F & q in lim_filter F holds p=q
  proof
    let T be Hausdorff non empty TopSpace,
    F be Filter of the carrier of T,
    p,q being Point of T such that
A1: p in lim_filter F and
A2: q in lim_filter F;
    consider p0 be Point of T such that
A3: p=p0 and
A4: F is_filter-finer_than NeighborhoodSystem p0 by A1;
    consider q0 be Point of T such that
A5: q=q0 and
A6: F is_filter-finer_than NeighborhoodSystem q0 by A2;
    now
      assume p<>q;
      then consider G1,G2 being Subset of T such that
A7:   G1 is open and
A8:   G2 is open and
A9:   p in G1 and
A10:  q in G2 and
A11:  G1 misses G2 by PRE_TOPC:def 10;
      G1 in NeighborhoodSystem p &
      G2 in NeighborhoodSystem q by A7,A8,A9,A10,CONNSP_2:3,YELLOW19:2;
      then {} in F by A3,A4,A5,A6,A11,CARD_FIL:def 1;
      hence contradiction by CARD_FIL:def 1;
    end;
    hence thesis;
  end;
