
theorem Th34:
  for T being non empty TopSpace
  for x,y being Element of InclPoset the topology of T st x is_way_below y
  for F being Subset-Family of T st F is open & y c= union F
  ex G being finite Subset of F st x c= union G
proof
  let T be non empty TopSpace;
  set L = InclPoset the topology of T;
  let x,y be Element of L such that
A1: x << y;
  let F be Subset-Family of T;
  assume that
A2: F is open and
A3: y c= union F;
  reconsider A = F as Subset of L by A2,YELLOW_1:25;
  sup A = union F by YELLOW_1:22;
  then y <= sup A by A3,YELLOW_1:3;
  then consider B being finite Subset of L such that
A4: B c= A and
A5: x <= sup B by A1,Th18;
  reconsider G = B as finite Subset of F by A4;
  take G;
  sup B = union G by YELLOW_1:22;
  hence thesis by A5,YELLOW_1:3;
end;
