
theorem Th5:
  for T being transitive lower non empty TopRelStr for A being
  Subset of T st A is open holds A is lower
proof
  let T be transitive lower non empty TopRelStr;
  reconsider BB = the set of all (uparrow x)` where x is Element of T as
  prebasis of T by Def1;
  let A be Subset of T such that
A1: A in the topology of T;
  let x,y be Element of T;
  consider F being Basis of T such that
A2: F c= FinMeetCl BB by CANTOR_1:def 4;
  the topology of T c= UniCl F by CANTOR_1:def 2;
  then consider Y being Subset-Family of T such that
A3: Y c= F and
A4: A = union Y by A1,CANTOR_1:def 1;
  assume x in A;
  then consider Z being set such that
A5: x in Z and
A6: Z in Y by A4,TARSKI:def 4;
  Z in F by A3,A6;
  then consider X being Subset-Family of T such that
A7: X c= BB and
  X is finite and
A8: Z = Intersect X by A2,CANTOR_1:def 3;
  assume
A9: x >= y;
  now
    let Q be set;
    assume
A10: Q in X;
    then Q in BB by A7;
    then consider z being Element of T such that
A11: Q = (uparrow z)`;
    (uparrow z) misses Q by A11,XBOOLE_1:79;
    then
A12: (uparrow z) /\ Q = {}T;
    x in Q by A5,A8,A10,SETFAM_1:43;
    then not x in uparrow z by A12,XBOOLE_0:def 4;
    then not x >= z by WAYBEL_0:18;
    then not y >= z by A9,ORDERS_2:3;
    then not y in uparrow z by WAYBEL_0:18;
    hence y in Q by A11,XBOOLE_0:def 5;
  end;
  then y in Z by A8,SETFAM_1:43;
  hence thesis by A4,A6,TARSKI:def 4;
end;
