
theorem
  for T being Scott TopAugmentation of BoolePoset {0} holds the topology
  of T = the topology of Sierpinski_Space
proof
  let T be Scott TopAugmentation of BoolePoset {0};
A1: LattPOSet BooleLatt {0} =
     RelStr(#the carrier of BooleLatt {0}, LattRel( BooleLatt {0})#)
     by LATTICE3:def 2;
A2: the RelStr of T = BoolePoset {0} by YELLOW_9:def 4;
  then
A3: the carrier of T = the carrier of LattPOSet BooleLatt {0}
            by YELLOW_1:def 2
    .= bool {0} by A1,LATTICE3:def 1
    .= {0,1} by CARD_1:49,ZFMISC_1:24;
  then reconsider j = {0}, z = 0 as Element of BoolePoset {0}
              by A2,TARSKI:def 2,CARD_1:49;
A4: now
    let x be object;
    assume x in the topology of Sierpinski_Space;
    then
A5: x in {{}, {{0}}, {0,{0}} } by Def9,CARD_1:49;
    per cases by A5,ENUMSET1:def 1,CARD_1:49;
    suppose
      x = {};
      hence x in the topology of T by PRE_TOPC:1;
    end;
    suppose
A6:   x = {{0}};
      then reconsider x9 = x as Subset of T by A3,ZFMISC_1:7,CARD_1:49;
      for a,b being Element of T st a in x9 & a <= b holds b in x9
      proof
        let a,b be Element of T;
        assume that
A7:     a in x9 and
A8:     a <= b;
A9:     a = {0} by A6,A7,TARSKI:def 1;
A10:    b <> 0
        proof
          assume
A11:      b = 0;
          [a, b] in the InternalRel of T by A8,ORDERS_2:def 5;
          then j <= z by A2,A9,A11,ORDERS_2:def 5;
          then {0} c= {} by YELLOW_1:2;
          hence thesis;
        end;
        b = 0 or b = 1 by A3,TARSKI:def 2;
        hence thesis by A6,A10,TARSKI:def 1,CARD_1:49;
      end;
      then
A12:  x9 is upper by WAYBEL_0:def 20;
      for D being non empty directed Subset of T st sup D in x9 holds D
      meets x9
      proof
        let D be non empty directed Subset of T;
        assume
A13:    sup D in x9;
        D <> {0}
        proof
          assume D = {0};
          then sup D = sup {z} by A2,YELLOW_0:17,26
            .= 0 by YELLOW_0:39;
          hence thesis by A6,A13,TARSKI:def 1;
        end;
        then D = {1} or D = {0,1} by A3,ZFMISC_1:36;
        then
A14:    1 in D by TARSKI:def 1,def 2;
        1 in x9 by A6,TARSKI:def 1,CARD_1:49;
        hence thesis by A14,XBOOLE_0:3;
      end;
      then x9 is inaccessible by WAYBEL11:def 1;
      then x9 is open by A12,WAYBEL11:def 4;
      hence x in the topology of T;
    end;
    suppose
      x = {0,1};
      hence x in the topology of T by A3,PRE_TOPC:def 1;
    end;
  end;
  reconsider c = {z} as Subset of T by A2;
  now
    set a = 0, b = {0};
    let y be object;
A15: not b in {z};
    a c= b & a in {z} by TARSKI:def 1;
    then not {z} is upper by A15,WAYBEL_7:7;
    then not c is upper by A2,WAYBEL_0:25;
    then
A16: not c is open by WAYBEL11:def 4;
    assume
A17: y in the topology of T;
    then reconsider x = y as Subset of T;
    x = {} or x = {0} or x = {1} or x = {0,1} by A3,ZFMISC_1:36;
    then y in {{}, {1}, {0,1} } by A17,A16,ENUMSET1:def 1;
    hence y in the topology of Sierpinski_Space by Def9;
  end;
  hence thesis by A4,TARSKI:2;
end;
