
theorem Th37:
  for T being injective T_0-TopSpace, S being Scott
  TopAugmentation of Omega T holds the TopStruct of S = the TopStruct of T
proof
  set SS = Sierpinski_Space, B = BoolePoset{0};
  let T be injective T_0-TopSpace, S be Scott TopAugmentation of Omega T;
  consider M being non empty set such that
A1: T is_Retract_of product (M --> SS) by WAYBEL18:19;
  consider c being continuous Function of T, product (M --> SS), r being
  continuous Function of product (M --> SS), T such that
A2: r * c = id T by A1;
A3: the TopStruct of T = the TopStruct of Omega T by Def2;
A4: the TopStruct of product (M --> SS) = the TopStruct of Omega product (M
  --> SS) by Def2;
  then reconsider
  c1a = c as Function of Omega T, Omega product (M --> SS) by A3;
  set S2M = the Scott TopAugmentation of product (M --> B);
A5: the TopStruct of S = the TopStruct of S;
A6: the RelStr of S2M = the RelStr of product (M --> B) by YELLOW_9:def 4;
  then reconsider c1 = c as Function of Omega T, product (M --> B) by A3,Th3;
A7: the RelStr of S = the RelStr of Omega T by YELLOW_9:def 4;
  then reconsider c2 = c1 as Function of S, S2M by A6;
A8: the carrier of S2M = the carrier of product (M --> SS) by Th3;
  then reconsider rr = r as Function of S2M, T;
A9: the topology of S2M = the topology of product (M --> SS) by WAYBEL18:15;
  then reconsider W = T as monotone-convergence non empty TopSpace by A1,A8
,Th36;
  Omega product (M --> SS) = Omega S2M by A9,A8,Th13;
  then
A10: the RelStr of Omega product (M --> SS) = the RelStr of product (M --> B
  ) by Th16
    .= the RelStr of S2M by YELLOW_9:def 4;
  reconsider r1 = r as Function of product (M --> B), Omega T by A8,A6,A3;
A11: the RelStr of Omega S2M = the RelStr of product (M --> B) by Th16;
  then reconsider rr1 = r1 as Function of Omega S2M, Omega T;
  reconsider r2 = r1 as Function of S2M, S by A6,A7;
  reconsider r3 = r2 as Function of product (M --> SS), S by Th3;
  the TopStruct of Omega S2M = the TopStruct of S2M by Def2;
  then rr1 is continuous by A9,A8,A3,YELLOW12:36;
  then r2 is directed-sups-preserving by A6,A7,A11,WAYBEL21:6;
  then r3 is continuous by A9,A8,A5,YELLOW12:36;
  then
A12: r3 * c is continuous;
  reconsider c1a as continuous Function of Omega W, Omega product (M --> SS)
  by A3,A4,YELLOW12:36;
  c2 = c1a;
  then
A13: c2 is directed-sups-preserving by A7,A10,WAYBEL21:6;
  the TopStruct of T = the TopStruct of T;
  then rr is continuous by A9,A8,YELLOW12:36;
  then rr * c2 is continuous by A13;
  hence thesis by A2,A12,YELLOW14:33;
end;
