
theorem Th13:
  for I being non empty set holds the set of all product ((Carrier (I -->
  Sierpinski_Space))+*(i,{1})) where i is Element of I is
  prebasis of product (I --> Sierpinski_Space)
proof
  let I be non empty set;
  set IS = I --> Sierpinski_Space, pB = the set of all
 product ((Carrier IS)+*(i,{1})) where i is Element of I;
  set P = product_prebasis IS;
A1: P is prebasis of product IS by Def3;
  then
A2: P c= the topology of product IS by TOPS_2:64;
  pB c= bool the carrier of product IS
  proof
    let x be object;
    reconsider xx=x as set by TARSKI:1;
    assume x in pB;
    then consider i being Element of I such that
A3: x = product ((Carrier IS)+*(i,{1}));
    product ((Carrier IS)+*(i,{1})) c= product Carrier IS
    proof
      let y be object;
      assume y in product ((Carrier IS)+*(i,{1}));
      then consider g being Function such that
A4:   y = g and
A5:   dom g = dom ((Carrier IS)+*(i,{1})) and
A6:   for z being object st z in dom ((Carrier IS)+*(i,{1})) holds g.z in
      ((Carrier IS)+*(i,{1})).z by CARD_3:def 5;
A7:   for z being object st z in dom Carrier IS
holds g.z in ( Carrier IS ) . z
      proof
        let z be object;
        assume
A8:     z in dom (Carrier IS);
        then
A9:     z in I;
        then consider R being 1-sorted such that
A10:    R = IS.z and
A11:    (Carrier IS).z = the carrier of R by PRALG_1:def 15;
A12:    the carrier of R = the carrier of Sierpinski_Space by A9,A10,FUNCOP_1:7
          .= {0,1} by Def9;
        z in dom ((Carrier IS)+*(i,{1})) by A8,FUNCT_7:30;
        then
A13:    g.z in ((Carrier IS)+*(i,{1})).z by A6;
        per cases;
        suppose
          z = i;
          then ((Carrier IS)+*(i,{1})).z = {1} by A8,FUNCT_7:31;
          then g.z = 1 by A13,TARSKI:def 1;
          hence thesis by A11,A12,TARSKI:def 2;
        end;
        suppose
          z <> i;
          hence thesis by A13,FUNCT_7:32;
        end;
      end;
      dom g = dom Carrier IS by A5,FUNCT_7:30;
      hence thesis by A4,A7,CARD_3:def 5;
    end;
    then xx c= the carrier of product IS by A3,Def3;
    hence thesis;
  end;
  then reconsider B = pB as Subset-Family of product IS;
  reconsider B as Subset-Family of product IS;
A14: B c= P
  proof
    {1} c= {0,1} by ZFMISC_1:7;
    then reconsider V = {1} as Subset of Sierpinski_Space by Def9;
    let x be object;
    assume
A15: x in B;
    then consider i being Element of I such that
A16: x = product ((Carrier IS)+*(i,{1}));
    reconsider y = x as Subset of product Carrier IS by A15,Def3;
    {1} in {{}, {1}, {0,1} } by ENUMSET1:def 1;
    then {1} in the topology of Sierpinski_Space by Def9;
    then
A17: V is open;
    Sierpinski_Space = IS.i & y = product ((Carrier IS) +* (i,V)) by A16;
    hence thesis by A17,Def2;
  end;
  reconsider P as Subset-Family of product IS by Def3;
  reconsider P as Subset-Family of product IS;
  FinMeetCl P is Basis of product IS by A1,YELLOW_9:23;
  then reconsider F = (FinMeetCl P) \ {{}} as Basis of product IS by Th2;
A18: F c= FinMeetCl B
  proof
    let x be object;
    assume
A19: x in F;
    then reconsider y = x as Subset of product IS;
    x in FinMeetCl P by A19,XBOOLE_0:def 5;
    then consider Y1 being Subset-Family of product IS such that
A20: Y1 c= P and
A21: Y1 is finite and
A22: y = Intersect Y1 by CANTOR_1:def 3;
    reconsider Y2 = Y1 /\ B as Subset-Family of product IS;
A23: Y2 c= B & Y2 is finite by A21,FINSET_1:1,XBOOLE_1:17;
A24: the carrier of product IS = product Carrier IS by Def3;
A25: not x in {{}} by A19,XBOOLE_0:def 5;
A26: Intersect Y2 c= Intersect Y1
    proof
      let z be object;
      assume
A27:  z in Intersect Y2;
      then
A28:  z in product Carrier IS by A24;
      for Y being set st Y in Y1 holds z in Y
      proof
        let Y be set;
        assume
A29:    Y in Y1;
        then reconsider Y9 = Y as Subset of product Carrier IS by Def3;
        consider i being set, S being TopStruct, V being Subset of S such that
A30:    i in I and
A31:    V is open and
A32:    S = IS.i and
A33:    Y9 = product ((Carrier IS) +* (i,V)) by A20,A29,Def2;
        reconsider V9 = V as Subset of Sierpinski_Space by A30,A32,FUNCOP_1:7;
        V in the topology of S by A31;
        then V9 in the topology of Sierpinski_Space by A30,A32,FUNCOP_1:7;
        then
A34:    V9 in {{}, {1}, {0,1} } by Def9;
A35:    i in dom (Carrier IS) by A30,PARTFUN1:def 2;
A36:    V9 <> {}
        proof
          ((Carrier IS)+*(i,{})).i = {} & i in dom ((Carrier IS)+*(i,{}))
          by A35,FUNCT_7:30,31;
          then
A37:      {} in rng ((Carrier IS)+*(i,{})) by FUNCT_1:def 3;
          assume V9 = {};
          then Y9 = {} by A33,A37,CARD_3:26;
          then y = {} by A22,A29,MSSUBFAM:3;
          hence thesis by A25,TARSKI:def 1;
        end;
        reconsider i9 = i as Element of I by A30;
A38:    ex RR being 1-sorted st RR = IS.i & (Carrier IS).i = the carrier
        of RR by A30,PRALG_1:def 15;
        per cases by A34,A36,ENUMSET1:def 1;
        suppose
          V9 = {1};
          then Y9 = product ((Carrier IS)+*(i9,{1})) by A33;
          then Y in B;
          then Y in Y2 by A29,XBOOLE_0:def 4;
          hence thesis by A27,SETFAM_1:43;
        end;
        suppose
          V9 = {0,1};
          then V9 = the carrier of Sierpinski_Space by Def9
            .= (Carrier IS).i by A30,A38,FUNCOP_1:7;
          hence thesis by A28,A33,FUNCT_7:35;
        end;
      end;
      hence thesis by A27,SETFAM_1:43;
    end;
    Intersect Y1 c= Intersect Y2 by SETFAM_1:44,XBOOLE_1:17;
    then y = Intersect Y2 by A22,A26;
    hence thesis by A23,CANTOR_1:def 3;
  end;
  pB c= the topology of product IS by A14,A2;
  hence thesis by A18,CANTOR_1:def 4,TOPS_2:64;
end;
