
theorem Th12:
  for S be lower-bounded sup-Semilattice for x be Element of
InclPoset Ids S holds x is compact iff ex a be Element of S st x = downarrow a
proof
  let S be lower-bounded sup-Semilattice;
  let x be Element of InclPoset Ids S;
  thus x is compact implies ex a be Element of S st x = downarrow a
  proof
    assume x is compact;
    then x is principal Ideal of S by Th11;
    hence thesis by WAYBEL_0:48;
  end;
  thus (ex a be Element of S st x = downarrow a) implies x is compact
  by WAYBEL_0:48,Th11;
end;
