reserve x,y,z,X for set,
  T for Universe;

theorem Th35:
  for T being non empty TopSpace holds T is Hausdorff iff for N
  being net of T, p,q being Point of T st p in Lim N & q in Lim N holds p = q
proof
  let T be non empty TopSpace;
  thus T is Hausdorff implies for N being net of T, p,q being Point of T st p
  in Lim N & q in Lim N holds p = q
  proof
    assume
A1: T is Hausdorff;
    let N be net of T;
    given p1,p2 being Point of T such that
A2: p1 in Lim N and
A3: p2 in Lim N and
A4: p1 <> p2;
    consider W,V being Subset of T such that
A5: W is open and
A6: V is open and
A7: p1 in W and
A8: p2 in V and
A9: W misses V by A1,A4;
    V is a_neighborhood of p2 by A6,A8,CONNSP_2:3;
    then
A10: N is_eventually_in V by A3,Def15;
    W is a_neighborhood of p1 by A5,A7,CONNSP_2:3;
    then N is_eventually_in W by A2,Def15;
    hence contradiction by A9,A10,Th17;
  end;
  assume
A11: for N being net of T, p,q being Point of T st p in Lim N & q in Lim
  N holds p = q;
  given p,q be Point of T such that
A12: p <> q and
A13: for W,V being Subset of T st W is open & V is open & p in W & q in
  V holds W meets V;
  set pN = [:OpenNeighborhoods p,OpenNeighborhoods q:];
  set cT = the carrier of T, cpN = the carrier of pN;
  deffunc F(Element of cpN) = $1`1 /\ $1`2;
A14: for i being Element of cpN holds cT meets F(i)
  proof
    let i be Element of cpN;
    consider W being Subset of T such that
A15: W = i`1 and
A16: p in W & W is open by Th29;
    consider V being Subset of T such that
A17: V = i`2 and
A18: q in V & V is open by Th29;
    i`1 meets i`2 by A13,A15,A16,A17,A18;
    then W /\ V c= cT & i`1 /\ i`2 <> {} by XBOOLE_0:def 7;
    then cT /\ (i`1 /\ i`2) <> {} by A15,A17,XBOOLE_1:28;
    hence thesis by XBOOLE_0:def 7;
  end;
  consider f being Function of cpN, cT such that
A19: for i being Element of cpN holds f.i in F(i) from FUNCT_2:sch 10(
  A14);
  reconsider N = NetStr(#the carrier of pN, the InternalRel of pN, f#) as net
  of T by Lm1,Lm2;
A20: cpN = [:the carrier of OpenNeighborhoods p, the carrier of
  OpenNeighborhoods q:] by YELLOW_3:def 2;
  now
    let V be a_neighborhood of q;
A21: N is_eventually_in Int V
    proof
A22:  [#]T in the carrier of OpenNeighborhoods p by Th30;
      q in Int V & Int V is open by CONNSP_2:def 1;
      then Int V in the carrier of OpenNeighborhoods q by Th30;
      then reconsider i = [[#]T,Int V] as Element of N by A20,A22,ZFMISC_1:87;
      take i;
      let j be Element of N;
      reconsider j9=j, i9=i as Element of pN;
      consider j1 being Element of OpenNeighborhoods p, j2 being Element of
      OpenNeighborhoods q such that
A23:  j = [j1,j2] by A20,DOMAIN_1:1;
A24:  j`2 = j2 by A23;
      consider W1 being Subset of T such that
A25:  j1 = W1 and
      p in W1 and
      W1 is open by Th29;
      consider W2 being Subset of T such that
A26:  j2 = W2 and
      q in W2 and
      W2 is open by Th29;
      assume i <= j;
      then [i,j] in the InternalRel of pN by ORDERS_2:def 5;
      then i9 <= j9 by ORDERS_2:def 5;
      then i9`2 = Int V & i9`2 <= j9`2 by YELLOW_3:12;
      then W2 c= Int V by A24,A26,Th31;
      then
A27:  W1 /\ W2 c= (Int V) /\ [#]T by XBOOLE_1:27;
      j`1 = j1 by A23;
      then f.j in W1 /\ W2 by A19,A24,A25,A26;
      then f.j in (Int V) /\ [#]T by A27;
      hence thesis by XBOOLE_1:28;
    end;
    Int V c= V by TOPS_1:16;
    hence N is_eventually_in V by A21,WAYBEL_0:8;
  end;
  then
A28: q in Lim N by Def15;
  now
    let V be a_neighborhood of p;
A29: N is_eventually_in Int V
    proof
A30:  [#]T in the carrier of OpenNeighborhoods q by Th30;
      p in Int V & Int V is open by CONNSP_2:def 1;
      then Int V in the carrier of OpenNeighborhoods p by Th30;
      then reconsider i = [Int V,[#]T] as Element of N by A20,A30,ZFMISC_1:87;
      take i;
      let j be Element of N;
      reconsider j9=j, i9=i as Element of pN;
      consider j1 being Element of OpenNeighborhoods p, j2 being Element of
      OpenNeighborhoods q such that
A31:  j = [j1,j2] by A20,DOMAIN_1:1;
A32:  j`1 = j1 by A31;
      consider W2 being Subset of T such that
A33:  j2 = W2 and
      q in W2 and
      W2 is open by Th29;
      consider W1 being Subset of T such that
A34:  j1 = W1 and
      p in W1 and
      W1 is open by Th29;
      assume i <= j;
      then [i,j] in the InternalRel of pN by ORDERS_2:def 5;
      then i9 <= j9 by ORDERS_2:def 5;
      then i9`1 = Int V & i9`1 <= j9`1 by YELLOW_3:12;
      then W1 c= Int V by A32,A34,Th31;
      then
A35:  W1 /\ W2 c= (Int V) /\ [#]T by XBOOLE_1:27;
      j`2 = j2 by A31;
      then f.j in W1 /\ W2 by A19,A32,A34,A33;
      then f.j in (Int V) /\ [#]T by A35;
      hence thesis by XBOOLE_1:28;
    end;
    Int V c= V by TOPS_1:16;
    hence N is_eventually_in V by A29,WAYBEL_0:8;
  end;
  then p in Lim N by Def15;
  hence contradiction by A11,A12,A28;
end;
