
theorem Th38:
  for T being non empty TopSpace
  for x,y being Element of InclPoset the topology of T
  st ex Z being Subset of T st x c= Z & Z c= y & Z is compact holds x << y
proof
  let T be non empty TopSpace;
  set L = InclPoset the topology of T;
  let x,y be Element of L;
  given Z be Subset of T such that
A1: x c= Z and
A2: Z c= y and
A3: Z is compact;
A4: L = RelStr(#the topology of T, RelIncl the topology of T#)
  by YELLOW_1:def 1;
  then
A5: x in the topology of T;
  y in the topology of T by A4;
  then reconsider X = x, Y = y as Subset of T by A5;
  let D be non empty directed Subset of L;
A6: sup D = union D by YELLOW_1:22;
  reconsider F = D as Subset-Family of T by A4,XBOOLE_1:1;
  reconsider F as Subset-Family of T;
A7: F is open
  proof
    let a be Subset of T;
    assume a in F;
    hence a in the topology of T by A4;
  end;
  assume y <= sup D;
  then Y c= union F by A6,YELLOW_1:3;
  then Z c= union F by A2;
  then F is Cover of Z by SETFAM_1:def 11;
  then consider G being Subset-Family of T such that
A8: G c= F and
A9: G is Cover of Z and
A10: G is finite by A3,A7;
  consider d being Element of L such that
A11: d in D and
A12: d is_>=_than G by A8,A10,WAYBEL_0:1;
  take d;
  thus d in D by A11;
A13: now
    let B be set;
    assume
A14: B in G;
    then B in D by A8;
    then reconsider e = B as Element of L;
    d >= e by A12,A14;
    hence B c= d by YELLOW_1:3;
  end;
A15: Z c= union G by A9,SETFAM_1:def 11;
  union G c= d by A13,ZFMISC_1:76;
  then Z c= d by A15;
  then X c= d by A1;
  hence thesis by YELLOW_1:3;
end;
