
theorem Th35:
  for T being lower up-complete TopLattice for A being Subset of T
  st A is open holds A is property(S)
proof
  let T be lower up-complete TopLattice;
  defpred P[Subset of T] means $1 is property(S);
A1: ex K being prebasis of T st for A being Subset of T st A in K holds P[A]
  proof
    reconsider K = the set of all (uparrow x)` where x is Element of T
    as prebasis of T by Def1;
    take K;
    let A be Subset of T;
    assume A in K;
    then consider x being Element of T such that
A2: A = (uparrow x)`;
    let D be non empty directed Subset of T;
    assume
A3: sup D in A;
    set y = the Element of D;
    reconsider y as Element of T;
    take y;
    thus y in D;
    let z be Element of T;
    assume that
A4: z in D and
    z >= y and
A5: not z in A;
A6: z >= x by A5,A2,YELLOW_9:1;
    not x <= sup D by A3,A2,YELLOW_9:1;
    then
A7: not z <= sup D by A6,ORDERS_2:3;
A8: ex_sup_of D,T by WAYBEL_0:75;
A9: ex_sup_of {z},T by YELLOW_0:38;
    {z} c= D by A4,ZFMISC_1:31;
    then sup {z} <= sup D by A8,A9,YELLOW_0:34;
    hence thesis by A7,YELLOW_0:39;
  end;
A10: for A1,A2 being Subset of T st P[A1] & P[A2] holds P[A1 /\ A2] by Lm3;
A11: P[[#]T] by Lm4;
A12: 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(A1,A12,A10,
  A11);
end;
