
theorem ::p.122 lemma 3.4.(i) part II
  for I being non empty set for T being Scott TopAugmentation of product
  (I --> BoolePoset{0}) holds T is injective
proof
  let I be non empty set, T be Scott TopAugmentation of product (I -->
  BoolePoset{0});
  set IB = I --> BoolePoset{0}, IS = I --> Sierpinski_Space;
A1: dom Carrier IB = I by PARTFUN1:def 2
    .= dom Carrier IS by PARTFUN1:def 2;
A2: the topology of T = the topology of product (I --> Sierpinski_Space) by
Th15;
  LattPOSet BooleLatt{0} = RelStr(#the carrier of BooleLatt{0}, LattRel(
    BooleLatt{0})#) by LATTICE3:def 2;
  then
A3: 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;
A4: 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
A5: i in I;
    then consider R1 being 1-sorted such that
A6: R1 = IB.i and
A7: (Carrier IB).i = the carrier of R1 by PRALG_1:def 15;
    consider R2 being 1-sorted such that
A8: R2 = IS.i and
A9: (Carrier IS).i = the carrier of R2 by A5,PRALG_1:def 15;
    the carrier of R1 = {0,1} by A3,A5,A6,FUNCOP_1:7
      .= the carrier of Sierpinski_Space by Def9
      .= the carrier of R2 by A5,A8,FUNCOP_1:7;
    hence thesis by A7,A9;
  end;
  for i being Element of I holds (I --> Sierpinski_Space).i is injective;
  then
A10: product (I --> Sierpinski_Space) is injective by Th7;
  the RelStr of T = product (I --> BoolePoset{0}) by YELLOW_9:def 4;
  then the carrier of T = product Carrier (I --> BoolePoset{0}) by
YELLOW_1:def 4
    .= product Carrier (I --> Sierpinski_Space) by A1,A4,FUNCT_1:2
    .= the carrier of product (I --> Sierpinski_Space) by Def3;
  hence thesis by A2,A10,Th16;
end;
