
theorem Th3: :: see WAYBEL18:16
  for I being non empty set, T being Scott TopAugmentation of
  product(I --> BoolePoset{0})
   holds the carrier of T = the carrier of product(I
  --> Sierpinski_Space)
proof
  set S = Sierpinski_Space, B = BoolePoset{0};
  let I be non empty set, T be Scott TopAugmentation of product(I -->
  BoolePoset{0});
A1: dom Carrier (I --> B) = I by PARTFUN1:def 2;
A2: dom Carrier (I --> S) = I by PARTFUN1:def 2;
A3: for x being object
   st x in dom Carrier (I --> S) holds (Carrier (I --> B)).
  x = (Carrier (I --> S)).x
  proof
    let x be object;
    assume
A4: x in dom Carrier (I --> S);
    then
A5: ex T being 1-sorted st T = (I --> S).x & (Carrier (I --> S)).x = the
    carrier of T by PRALG_1:def 15;
    ex R being 1-sorted st R = (I --> B).x & (Carrier (I --> B)).x = the
    carrier of R by A4,PRALG_1:def 15;
    hence (Carrier (I --> B)).x = the carrier of B by A4,FUNCOP_1:7
      .= bool {0} by WAYBEL_7:2
      .= the carrier of S by WAYBEL18:def 9,YELLOW14:1
      .= (Carrier (I --> S)).x by A4,A5,FUNCOP_1:7;
  end;
  the RelStr of T = the RelStr of product(I --> B) by YELLOW_9:def 4;
  hence the carrier of T = product Carrier (I --> B) by YELLOW_1:def 4
    .= product Carrier (I --> S) by A1,A2,A3,FUNCT_1:2
    .= the carrier of product(I --> S) by WAYBEL18:def 3;
end;
