
theorem
  for T being non empty TopSpace for B being prebasis of T for x,y being
  Element of InclPoset the topology of T st x c= y holds x << y iff for F being
  Subset of B st y c= union F ex G being finite Subset of F st x c= union G
proof
  let T be non empty TopSpace, B be prebasis of T;
  consider BB being Basis of T such that
A1: BB c= FinMeetCl B by CANTOR_1:def 4;
  set BT = BoolePoset the carrier of T;
  set L = InclPoset the topology of T;
  let x,y be Element of L such that
A2: x c= y;
A3: B c= the topology of T by TOPS_2:64;
  hereby
    assume
A4: x << y;
    let F be Subset of B such that
A5: y c= union F;
    reconsider FF = F as Subset-Family of T by XBOOLE_1:1;
    FF is open
    proof
      let a be Subset of T;
      assume a in FF;
      then a in B;
      hence a in the topology of T by A3;
    end;
    hence ex G being finite Subset of F st x c= union G by A4,A5,WAYBEL_3:34;
  end;
  BT = InclPoset bool the carrier of T by YELLOW_1:4;
  then
A6: BT = RelStr(#bool the carrier of T, RelIncl bool the carrier of T#) by
YELLOW_1:def 1;
  L = RelStr(#the topology of T, RelIncl the topology of T#) by YELLOW_1:def 1;
  then x in the topology of T & y in the topology of T;
  then reconsider X = x, Y = y as Subset of T;
  assume
A7: for F being Subset of B st y c= union F ex G being finite Subset of
  F st x c= union G;
A8: the topology of T c= UniCl BB by CANTOR_1:def 2;
  now
    deffunc F(set) = x\$1;
    let F be ultra Filter of BoolePoset the carrier of T such that
A9: x in F and
A10: not ex p being Element of T st p in y & p is_a_convergence_point_of F,T;
    defpred P[object,object] means
       ex A being set st A = $2 & $1 in A & not $2 in F;
A11: now
      let p be object;
      assume
A12:  p in y;
      then p in Y;
      then reconsider q = p as Element of T;
      not q is_a_convergence_point_of F,T by A10,A12;
      then consider A being Subset of T such that
A13:  A is open and
A14:  q in A and
A15:  not A in F;
      A in the topology of T by A13;
      then consider AY being Subset-Family of T such that
A16:  AY c= BB and
A17:  A = union AY by A8,CANTOR_1:def 1;
      consider ay being set such that
A18:  q in ay and
A19:  ay in AY by A14,A17,TARSKI:def 4;
      reconsider ay as Subset of T by A19;
      ay in BB by A16,A19;
      then consider BY being Subset-Family of T such that
A20:  BY c= B and
A21:  BY is finite and
A22:  ay = Intersect BY by A1,CANTOR_1:def 3;
      ay c= A by A17,A19,ZFMISC_1:74;
      then not ay in F by A15,Th7;
      then BY is not Subset of F by A21,A22,Th11;
      then consider r being object such that
A23:  r in BY & not r in F by TARSKI:def 3;
       reconsider A=r as set by TARSKI:1;
      take r;
      thus r in B by A20,A23;
      thus P[p,r]
       proof
        take A;
        thus A = r & p in A & not r in F by A18,A22,A23,SETFAM_1:43;
       end;
    end;
    consider f being Function such that
A24: dom f = y & rng f c= B and
A25: for p being object st p in y holds P[p,f.p] from FUNCT_1:sch 6(A11);
    reconsider FF = rng f as Subset of B by A24;
    y c= union FF
    proof
      let p be object;
      assume
A26:     p in y;
      then consider A being set such that
A27:     A = f.p & p in A & not f.p in F by A25;
       f.p in FF & p in f.p by A24,FUNCT_1:def 3,A27,A26;
      hence thesis by TARSKI:def 4;
    end;
    then consider G being finite Subset of FF such that
A28: x c= union G by A7;
    set gg = the Element of G;
    consider g being Function such that
A29: dom g = G & for z being set st z in G holds g.z = F(z) from
    FUNCT_1:sch 5;
A30: rng g c= F
    proof
      let a be object;
A31:  F is prime by Th22;
      assume a in rng g;
      then consider b being object such that
A32:  b in G and
A33:  a = g.b by A29,FUNCT_1:def 3;
      b in FF by A32;
      then b in B;
      then reconsider b as Subset of T;
      consider p being object such that
A34:     p in y & b = f.p by A24,A32,FUNCT_1:def 3;
      P[p,f.p] by A25,A34;
      then not b in F by A34;
      then
A35:  (the carrier of T)\b in F by A31,Th21;
      a = x\b by A29,A32,A33
        .= X /\ b` by SUBSET_1:13
        .= x /\ ((the carrier of T)\b);
      hence thesis by A9,A35,Th9;
    end;
    then reconsider GG = rng g as finite Subset-Family of T by A6,A29,
FINSET_1:8,XBOOLE_1:1;
    x <> Bottom BoolePoset the carrier of T by A9,Th4;
    then x <> {} by YELLOW_1:18;
    then
A36: G <> {} by A28,ZFMISC_1:2;
    now
      let a be object;
      assume
A37:  a in Intersect GG;
      now
        let z be set;
        assume z in G;
        then g.z in GG & g.z = x\z by A29,FUNCT_1:def 3;
        then a in x\z by A37,SETFAM_1:43;
        hence not a in z by XBOOLE_0:def 5;
      end;
      then not ex z being set st a in z & z in G;
      then
A38:  not a in x by A28,TARSKI:def 4;
      g.gg in GG & g.gg = x\gg by A36,A29,FUNCT_1:def 3;
      then a in x\gg by A37,SETFAM_1:43;
      hence contradiction by A38;
    end;
    then
A39: Intersect GG = {} by XBOOLE_0:def 1;
    Intersect GG in F by A30,Th11;
    then Bottom BoolePoset the carrier of T in F by A39,YELLOW_1:18;
    hence contradiction by Th4;
  end;
  hence thesis by A2,Th30;
end;
