
theorem Th29:
  for X be set for x be Element of BoolePoset X holds compactbelow
  x = the set of all y where y is finite Subset of x
proof
  let X be set;
  let x be Element of BoolePoset X;
  now
    let z be object;
    assume z in the set of all y where y is finite Subset of x ;
    then consider z9 be finite Subset of x such that
A1: z9 = z;
    x c= X by Th26;
    then z9 c= X;
    then reconsider z1 = z9 as Element of BoolePoset X by Th26;
    z1 is compact & z1 <= x by Th28,YELLOW_1:2;
    hence z in compactbelow x by A1;
  end;
  then
A2: the set of all y where y is finite Subset of x  c= compactbelow x;
  now
    let z be object;
    assume z in compactbelow x;
    then consider z9 be Element of BoolePoset X such that
A3: z9 = z and
A4: x >= z9 & z9 is compact;
    z9 is finite & z9 c= x by A4,Th28,YELLOW_1:2;
    hence z in the set of all y where y is finite Subset of x  by A3;
  end;
  then
  compactbelow x c= the set of all y where y is finite Subset of x ;
  hence thesis by A2,XBOOLE_0:def 10;
end;
