
theorem Th36:

:: Remark after 1.5. p. 144
  for T being Lawson complete TopLattice for A being Subset of T
  st A is open holds A is property(S)
proof
  let T be Lawson complete TopLattice;
  defpred P[Subset of T] means $1 is property(S);

set S = the Scott TopAugmentation of T,R = the lower correct TopAugmentation of
T;
A1: the RelStr of R = the RelStr of T by YELLOW_9:def 4;
A2: the RelStr of S = the RelStr of T by YELLOW_9:def 4;
A3: ex K being prebasis of T st for A being Subset of T st A in K holds P[A]
  proof
    reconsider K = (sigma T) \/ omega T as prebasis of T by Def3;
    take K;
    let A be Subset of T;
    reconsider AS = A as Subset of S by A2;
    reconsider AR = A as Subset of R by A1;
    assume A in K;
    then
    A in sigma T & sigma T = the topology of S or A in omega T & omega T =
    the topology of R by Def2,XBOOLE_0:def 3,YELLOW_9:51;
    then AS is open or AR is open;
    then AS is property(S) or AR is property(S) by Th35,WAYBEL11:15;
    hence thesis by A2,A1,YELLOW12:19;
  end;
A4: P[[#]T];
A5: for A1,A2 being Subset of T st P[A1] & P[A2] holds P[A1 /\ A2] by Lm3;
A6: for F being Subset-Family of T st for A being Subset of T st A in F
  holds P[A] holds P[union F] by Lm2;
  thus for A being Subset of T st A is open holds P[A] from TopInd(A3,A6,A5,
  A4);
end;
