
theorem Th22:
  for L being TopLattice, x being Element of L st for X being
  Subset of L st X is open holds X is upper holds uparrow x is compact
proof
  let L be TopLattice, x be Element of L such that
A1: for X being Subset of L st X is open holds X is upper;
  set P = uparrow x;
  let F be Subset-Family of L such that
A2: F is Cover of P and
A3: F is open;
  x <= x;
  then
A4: x in P by WAYBEL_0:18;
  P c= union F by A2,SETFAM_1:def 11;
  then consider Y being set such that
A5: x in Y and
A6: Y in F by A4,TARSKI:def 4;
  reconsider Y as Subset of L by A6;
  reconsider G = {Y} as Subset-Family of L;
  reconsider G as Subset-Family of L;
  take G;
  thus G c= F by A6,ZFMISC_1:31;
  Y is open by A3,A6;
  then Y is upper by A1;
  then P c= Y by A5,WAYBEL11:42;
  hence P c= union G by ZFMISC_1:25;
  thus thesis;
end;
