
theorem Th15: ::p.122 lemma 3.4.(i)
  for I being non empty set for T being Scott TopAugmentation of
product (I --> BoolePoset {0})
   holds the topology of T = the topology of product
  (I --> Sierpinski_Space)
proof
  let I be non empty set;
  set IB = I --> BoolePoset {0}, IS = I --> Sierpinski_Space;
A1: the carrier of product IB = product Carrier IB by YELLOW_1:def 4;
  LattPOSet BooleLatt{0} = RelStr(#the carrier of BooleLatt{0}, LattRel(
    BooleLatt{0})#) by LATTICE3:def 2;
  then
A2: the carrier of BoolePoset{0} = the carrier of BooleLatt{0}
              by YELLOW_1:def 2
    .= bool {{}} by LATTICE3:def 1
    .= {0,1} by CARD_1:49,ZFMISC_1:24;
A3: for i being object st i in dom Carrier IB holds (Carrier IB).i = (Carrier
  IS).i
  proof
    let i be object;
    assume i in dom Carrier IB;
    then
A4: i in I;
    then consider R1 being 1-sorted such that
A5: R1 = IB.i and
A6: (Carrier IB).i = the carrier of R1 by PRALG_1:def 15;
    consider R2 being 1-sorted such that
A7: R2 = IS.i and
A8: (Carrier IS).i = the carrier of R2 by A4,PRALG_1:def 15;
    the carrier of R1 = {0,1} by A2,A4,A5,FUNCOP_1:7
      .= the carrier of Sierpinski_Space by Def9
      .= the carrier of R2 by A4,A7,FUNCOP_1:7;
    hence thesis by A6,A8;
  end;
  reconsider P = the set of all
product ((Carrier IS)+*(ii,{1})) where ii is Element of I
 as prebasis of product (I --> Sierpinski_Space) by Th13;
  let T be Scott TopAugmentation of product (I --> BoolePoset {0});
A9: dom Carrier IB = I by PARTFUN1:def 2
    .= dom Carrier IS by PARTFUN1:def 2;
  reconsider T9 = T as complete Scott TopLattice;
A10: the RelStr of T = product (I --> BoolePoset {0}) by YELLOW_9:def 4;
  then T9 is algebraic by WAYBEL_8:17;
  then consider R being Basis of T9 such that
A11: R = {uparrow yy where yy is Element of T9 : yy in the carrier of
  CompactSublatt T9} by WAYBEL14:42;
A12: the carrier of T = product Carrier (I --> BoolePoset {0}) by A10,
YELLOW_1:def 4
    .= product Carrier (I --> Sierpinski_Space) by A9,A3,FUNCT_1:2
    .= the carrier of product (I --> Sierpinski_Space) by Def3;
  then reconsider P9 = P as Subset-Family of T;
  consider f being Function of BoolePoset I, product IB such that
A13: f is isomorphic and
A14: for Y being Subset of I holds f.Y = chi(Y,I) by Th14;
A15: Carrier IB = Carrier IS by A9,A3,FUNCT_1:2;
A16: FinMeetCl P c= R
  proof
    deffunc F(object) = product ((Carrier IS)+*($1,{1}));
    let X be object;
    consider F being Function such that
A17: dom F = I and
A18: for e being object st e in I holds F.e = F(e) from FUNCT_1:sch 3;
    assume
A19: X in FinMeetCl P;
    then reconsider X9 = X as Subset of product IS;
    consider ZZ being Subset-Family of product IS such that
A20: ZZ c= P and
A21: ZZ is finite and
A22: X = Intersect ZZ by A19,CANTOR_1:def 3;
    P c= rng F
    proof
      let w be object;
      assume w in P;
      then consider e being Element of I such that
A23:  w = product ((Carrier IS)+*(e,{1}));
      w = F.e by A18,A23;
      hence thesis by A17,FUNCT_1:def 3;
    end;
    then ZZ c= rng F by A20;
    then consider x9 being set such that
A24: x9 c= dom F and
A25: x9 is finite and
A26: F.:x9 = ZZ by A21,ORDERS_1:85;
    reconsider x9 as Subset of I by A17,A24;
    reconsider x = x9 as Element of BoolePoset I by WAYBEL_8:26;
    reconsider y = f.x as Element of product IB;
    set PP = {product ((Carrier IS)+*(i,{1})) where i is Element of I: i in x9
    };
A27: ZZ c= PP
    proof
      let w be object;
      assume w in ZZ;
      then consider e being object such that
A28:  e in dom F and
A29:  e in x9 and
A30:  w = F.e by A26,FUNCT_1:def 6;
      reconsider e as Element of I by A17,A28;
      w = product ((Carrier IS)+*(e,{1})) by A18,A30;
      hence thesis by A29;
    end;
    PP c= ZZ
    proof
      let w be object;
      assume w in PP;
      then consider e being Element of I such that
A31:  w = product ((Carrier IS)+*(e,{1})) and
A32:  e in x9;
      w = F.e by A18,A31;
      hence thesis by A17,A26,A32,FUNCT_1:def 6;
    end;
    then
A33: ZZ = PP by A27;
A34: uparrow y c= X9
    proof
      let z be object;
      assume
A35:  z in uparrow y;
      then reconsider z9 = z as Element of product IB;
      y <= z9 by A35,WAYBEL_0:18;
      then consider h1,h2 being Function such that
A36:  h1 = y and
A37:  h2 = z9 and
A38:  for i be object st i in I ex R being RelStr, xi,yi being Element
      of R st R = IB.i & xi = h1.i & yi = h2.i & xi <= yi by A1,YELLOW_1:def 4;
      z in the carrier of product IB by A35;
      then z in product Carrier IB by YELLOW_1:def 4;
      then
A39:  ex gg being Function st z = gg & dom gg = dom (Carrier IB) & for w
being object st w in dom (Carrier IB) holds gg.w in (Carrier IB).w
by CARD_3:def 5;
A40:  h1 = chi(x,I) by A14,A36;
      for Z being set st Z in ZZ holds z in Z
      proof
        let Z be set;
        assume Z in ZZ;
        then consider j being Element of I such that
A41:    Z = product ((Carrier IS)+*(j,{1})) and
A42:    j in x by A33;
        consider RS being RelStr, xj,yj being Element of RS such that
A43:    RS = IB.j and
A44:    xj = h1.j and
A45:    yj = h2.j and
A46:    xj <= yj by A38;
A47:    RS = BoolePoset {0} by A43;
A48:    xj = 1 by A40,A42,A44,FUNCT_3:def 3;
A49:    yj <> 0
        proof
          assume yj = 0;
          then {0} c= 0 by A46,A48,A47,YELLOW_1:2,CARD_1:49;
          hence thesis;
        end;
        reconsider b = yj as Element of BoolePoset {0} by A43;
A50:    dom ((Carrier IS)+*(j,{1})) = dom (Carrier IS) by FUNCT_7:30
          .= I by PARTFUN1:def 2;
A51:    b in the carrier of BoolePoset {0};
A52:    for w being object st w in dom ((Carrier IS)+*(j,{1})) holds h2.w
        in ((Carrier IS)+*(j,{1})).w
        proof
          let w be object;
          assume w in dom ((Carrier IS)+*(j,{1}));
          then
A53:      w in dom Carrier IS by A50,PARTFUN1:def 2;
          per cases;
          suppose
            w = j;
            then ((Carrier IS)+*(j,{1})).w = {1} & h2.w = 1 by A2,A45,A51,A49
,A53,FUNCT_7:31,TARSKI:def 2;
            hence thesis by TARSKI:def 1;
          end;
          suppose
            w <> j;
            then ((Carrier IS)+*(j,{1})).w =(Carrier IS).w by FUNCT_7:32;
            hence thesis by A15,A39,A37,A53;
          end;
        end;
        dom h2 = dom ((Carrier IS)+*(j,{1})) by A39,A37,A50,PARTFUN1:def 2;
        hence thesis by A37,A41,A52,CARD_3:def 5;
      end;
      hence thesis by A10,A12,A22,A35,SETFAM_1:43;
    end;
A54: X9 c= uparrow y
    proof
      set h1 = chi(x,I);
      let z be object;
      assume
A55:  z in X9;
      then reconsider z9 = z as Element of product IB by A10,A12;
      z in the carrier of product IB by A10,A12,A55;
      then z in product Carrier IB by YELLOW_1:def 4;
      then consider h2 being Function such that
A56:  z = h2 and
      dom h2 = dom (Carrier IB) and
A57:  for w being object st w in dom (Carrier IB) holds h2.w in (
      Carrier IB).w by CARD_3:def 5;
A58:  for i be object st i in I ex R being RelStr, xi,yi being Element of R
      st R = IB.i & xi = h1.i & yi = h2.i & xi <= yi
      proof
        let i be object;
        assume
A59:    i in I;
        then reconsider i9 = i as Element of I;
        ex RB being 1-sorted st RB = IB.i & (Carrier IB).i = the carrier
        of RB by A59,PRALG_1:def 15;
        then
A60:    (Carrier IB).i = {0,1} by A2,A59,FUNCOP_1:7;
        take RS = BoolePoset {0};
A61:    i in dom (Carrier IB) by A59,PARTFUN1:def 2;
        then
A62:    h2.i in (Carrier IB).i by A57;
        per cases;
        suppose
A63:      i in x;
          reconsider xi = 1, yi = 1 as Element of RS by A2,TARSKI:def 2;
          take xi,yi;
          thus RS = IB.i by A59,FUNCOP_1:7;
          thus xi = h1.i by A63,FUNCT_3:def 3;
A64:      ((Carrier IS)+*(i9,{1})).i9 = {1} by A9,A61,FUNCT_7:31;
          product ((Carrier IS)+*(i9,{1})) in ZZ by A33,A63;
          then z in product ((Carrier IS)+*(i9,{1})) by A22,A55,SETFAM_1:43;
          then consider h29 being Function such that
A65:      z = h29 and
          dom h29 = dom ((Carrier IS)+*(i9,{1})) and
A66:      for w being object st w in dom ((Carrier IS)+*(i9,{1})) holds
          h29.w in ((Carrier IS)+*(i9,{1})).w by CARD_3:def 5;
          i9 in dom ((Carrier IS)+*(i9,{1})) by A9,A61,FUNCT_7:30;
          then h29.i9 in ((Carrier IS)+*(i9,{1})).i9 by A66;
          hence yi = h2.i by A56,A65,A64,TARSKI:def 1;
          thus xi <= yi;
        end;
        suppose
A67:      not i in x;
          thus thesis
          proof
            per cases by A60,A62,TARSKI:def 2;
            suppose
A68:          h2.i = 0;
              reconsider xi = 0, yi = 0 as Element of RS by A2,TARSKI:def 2;
              take xi,yi;
              thus RS = IB.i by A59,FUNCOP_1:7;
              thus xi = h1.i by A59,A67,FUNCT_3:def 3;
              thus yi = h2.i by A68;
              thus xi <= yi;
            end;
            suppose
A69:          h2.i = 1;
              reconsider xi = 0, yi = 1 as Element of RS by A2,TARSKI:def 2;
              take xi,yi;
              thus RS = IB.i by A59,FUNCOP_1:7;
              thus xi = h1.i by A59,A67,FUNCT_3:def 3;
              thus yi = h2.i by A69;
              xi c= yi;
              hence xi <= yi by YELLOW_1:2;
            end;
          end;
        end;
      end;
      h1 = y by A14;
      then y <= z9 by A1,A56,A58,YELLOW_1:def 4;
      hence thesis by WAYBEL_0:18;
    end;
    reconsider Y = y as Element of T9 by A10;
    x is compact by A25,WAYBEL_8:28;
    then y is compact by A13,WAYBEL13:28;
    then Y is compact by A10,WAYBEL_8:9;
    then
A70: Y in the carrier of CompactSublatt T9 by WAYBEL_8:def 1;
    uparrow Y = uparrow {Y} by WAYBEL_0:def 18
      .= uparrow {y} by A10,WAYBEL_0:13
      .= uparrow y by WAYBEL_0:def 18;
    then X = uparrow Y by A54,A34,XBOOLE_0:def 10;
    hence thesis by A11,A70;
  end;
A71: rng f = the carrier of product IB by A13,WAYBEL_0:66;
  R c= FinMeetCl P
  proof
    deffunc F(Element of I) = product ((Carrier IS)+*($1,{1}));
    let X be object;
    assume
A72: X in R;
    then consider Y being Element of T9 such that
A73: X = uparrow Y and
A74: Y in the carrier of CompactSublatt T9 by A11;
    reconsider X9 = X as Subset of product IS by A12,A72;
    reconsider y = Y as Element of product IB by A10;
    consider x being object such that
A75: x in dom f and
A76: y = f.x by A71,FUNCT_1:def 3;
    reconsider x as Element of BoolePoset I by A75;
    Y is compact by A74,WAYBEL_8:def 1;
    then y is compact by A10,WAYBEL_8:9;
    then x is compact by A13,A76,WAYBEL13:28;
    then
A77: x is finite by WAYBEL_8:28;
A78: {F(jj) where jj is Element of I: jj in x} is finite from FRAENKEL:sch
    21(A77);
    set ZZ = {product ((Carrier IS)+*(jj,{1})) where jj is Element of I: jj in
    x };
    reconsider x9 = x as Subset of I by WAYBEL_8:26;
A79: ZZ c= P
    proof
      let z be object;
      assume z in ZZ;
      then ex j being Element of I st z = product ((Carrier IS)+*(j, {1})) & j
      in x9;
      hence thesis;
    end;
    then reconsider ZZ as Subset-Family of product IS by XBOOLE_1:1;
A80: uparrow Y = uparrow {Y} by WAYBEL_0:def 18
      .= uparrow {y} by A10,WAYBEL_0:13
      .= uparrow y by WAYBEL_0:def 18;
A81: Intersect ZZ c= X9
    proof
      set h1 = chi(x,I);
      let z be object;
      assume
A82:  z in Intersect ZZ;
      then reconsider z9 = z as Element of product IB by A10,A12;
      z in the carrier of product IB by A10,A12,A82;
      then z in product Carrier IB by YELLOW_1:def 4;
      then consider h2 being Function such that
A83:  z = h2 and
      dom h2 = dom (Carrier IB) and
A84:  for w being object st w in dom (Carrier IB) holds h2.w in (Carrier
      IB).w by CARD_3:def 5;
A85:  for i be object st i in I ex R being RelStr, xi,yi being Element of R
      st R = IB.i & xi = h1.i & yi = h2.i & xi <= yi
      proof
        let i be object;
        assume
A86:    i in I;
        then reconsider i9 = i as Element of I;
        ex RB being 1-sorted st RB = IB.i & (Carrier IB).i = the carrier
        of RB by A86,PRALG_1:def 15;
        then
A87:    (Carrier IB).i = {0,1} by A2,A86,FUNCOP_1:7;
        take RS = BoolePoset {0};
A88:    i in dom (Carrier IB) by A86,PARTFUN1:def 2;
        then
A89:    h2.i in (Carrier IB).i by A84;
        per cases;
        suppose
A90:      i in x;
          reconsider xi = 1, yi = 1 as Element of RS by A2,TARSKI:def 2;
          take xi,yi;
          thus RS = IB.i by A86,FUNCOP_1:7;
          thus xi = h1.i by A86,A90,FUNCT_3:def 3;
A91:      ((Carrier IS)+*(i9,{1})).i9 = {1} by A9,A88,FUNCT_7:31;
          product ((Carrier IS)+*(i9,{1})) in ZZ by A90;
          then z in product ((Carrier IS)+*(i9,{1})) by A82,SETFAM_1:43;
          then consider h29 being Function such that
A92:      z = h29 and
          dom h29 = dom ((Carrier IS)+*(i9,{1})) and
A93:      for w being object st w in dom ((Carrier IS)+*(i9,{1})) holds
          h29.w in ((Carrier IS)+*(i9,{1})).w by CARD_3:def 5;
          i9 in dom ((Carrier IS)+*(i9,{1})) by A9,A88,FUNCT_7:30;
          then h29.i9 in ((Carrier IS)+*(i9,{1})).i9 by A93;
          hence yi = h2.i by A83,A92,A91,TARSKI:def 1;
          thus xi <= yi;
        end;
        suppose
A94:      not i in x;
          thus thesis
          proof
            per cases by A87,A89,TARSKI:def 2;
            suppose
A95:          h2.i = 0;
              reconsider xi = 0, yi = 0 as Element of RS by A2,TARSKI:def 2;
              take xi,yi;
              thus RS = IB.i by A86,FUNCOP_1:7;
              thus xi = h1.i by A86,A94,FUNCT_3:def 3;
              thus yi = h2.i by A95;
              thus xi <= yi;
            end;
            suppose
A96:          h2.i = 1;
              reconsider xi = 0, yi = 1 as Element of RS by A2,TARSKI:def 2;
              take xi,yi;
              thus RS = IB.i by A86,FUNCOP_1:7;
              thus xi = h1.i by A86,A94,FUNCT_3:def 3;
              thus yi = h2.i by A96;
              xi c= yi;
              hence xi <= yi by YELLOW_1:2;
            end;
          end;
        end;
      end;
      h1 = f.x9 by A14
        .= y by A76;
      then y <= z9 by A1,A83,A85,YELLOW_1:def 4;
      hence thesis by A73,A80,WAYBEL_0:18;
    end;
    X9 c= Intersect ZZ
    proof
      let z be object;
      assume
A97:  z in X9;
      then reconsider z9 = z as Element of product IB by A10,A12;
      y <= z9 by A73,A80,A97,WAYBEL_0:18;
      then consider h1,h2 being Function such that
A98:  h1 = y and
A99:  h2 = z9 and
A100: for i be object st i in I ex R being RelStr, xi,yi being Element
      of R st R = IB.i & xi = h1.i & yi = h2.i & xi <= yi by A1,YELLOW_1:def 4;
      z in the carrier of product IB by A10,A12,A97;
      then z in product Carrier IB by YELLOW_1:def 4;
      then
A101: ex gg being Function st z = gg & dom gg = dom (Carrier IB) & for w
being object st w in dom (Carrier IB) holds gg.w in (Carrier IB).w
by CARD_3:def 5;
A102: h1 = f.x9 by A76,A98
        .= chi(x,I) by A14;
      for Z being set st Z in ZZ holds z in Z
      proof
        let Z be set;
        assume Z in ZZ;
        then consider j being Element of I such that
A103:   Z = product ((Carrier IS)+*(j,{1})) and
A104:   j in x;
        consider RS being RelStr, xj,yj being Element of RS such that
A105:   RS = IB.j and
A106:   xj = h1.j and
A107:   yj = h2.j and
A108:   xj <= yj by A100;
        reconsider a = xj, b = yj as Element of BoolePoset{0}
       by A105;
A109:   a <= b by A105,A108;
A110:   xj = 1 by A102,A104,A106,FUNCT_3:def 3;
A111:   yj <> 0
        proof
          assume yj = 0;
          then {0} c= 0 by A110,A109,YELLOW_1:2,CARD_1:49;
          hence thesis;
        end;
A112:   dom ((Carrier IS)+*(j,{1})) = dom (Carrier IS) by FUNCT_7:30
          .= I by PARTFUN1:def 2;
A113:   b in the carrier of BoolePoset{0};
A114:   for w being object st w in dom ((Carrier IS)+*(j,{1})) holds h2.w in
        ((Carrier IS)+*(j,{1})).w
        proof
          let w be object;
          assume w in dom ((Carrier IS)+*(j,{1}));
          then
A115:     w in dom Carrier IS by A112,PARTFUN1:def 2;
          per cases;
          suppose
            w = j;
            then
            ((Carrier IS)+*(j,{1})).w = {1} & h2.w = 1 by A2,A107,A113,A111
,A115,FUNCT_7:31,TARSKI:def 2;
            hence thesis by TARSKI:def 1;
          end;
          suppose
            w <> j;
            then ((Carrier IS)+*(j,{1})).w =(Carrier IS).w by FUNCT_7:32;
            hence thesis by A15,A101,A99,A115;
          end;
        end;
        dom h2 = dom ((Carrier IS)+*(j,{1})) by A101,A99,A112,PARTFUN1:def 2;
        hence thesis by A99,A103,A114,CARD_3:def 5;
      end;
      hence thesis by A97,SETFAM_1:43;
    end;
    then X9 = Intersect ZZ by A81;
    hence thesis by A79,A78,CANTOR_1:def 3;
  end;
  then R = FinMeetCl P by A16;
  then P9 is prebasis of T by A12,YELLOW_9:23;
  hence thesis by A12,YELLOW_9:26;
end;
