
theorem Th68: :: PROPOSITION 4.3.(vi)
  for L be lower-bounded continuous sup-Semilattice for B be
  with_bottom CLbasis of L holds the carrier of CompactSublatt InclPoset(Ids
  subrelstr B) = the set of all  downarrow b where b is Element of subrelstr B
proof
  let L be lower-bounded continuous sup-Semilattice;
  let B be with_bottom CLbasis of L;
  thus the carrier of CompactSublatt InclPoset(Ids subrelstr B) c= the set of
all  downarrow
  b where b is Element of subrelstr B
  proof
    let x be object;
    assume
A1: x in the carrier of CompactSublatt InclPoset(Ids subrelstr B);
    the carrier of CompactSublatt InclPoset(Ids subrelstr B) c= the
    carrier of InclPoset(Ids subrelstr B) by YELLOW_0:def 13;
    then reconsider x1 = x as Element of InclPoset(Ids subrelstr B) by A1;
    x1 is compact by A1,WAYBEL_8:def 1;
    then ex b be Element of subrelstr B st x1 = downarrow b by WAYBEL13:12;
    hence thesis;
  end;
  thus the set of all  downarrow b where b is Element of subrelstr B
  c= the carrier of CompactSublatt InclPoset(Ids subrelstr B)
  proof
    let x be object;
    assume
    x in the set of all  downarrow b where b is Element of subrelstr B ;
    then
A2: ex b be Element of subrelstr B st x = downarrow b;
    then reconsider x1 = x as Element of InclPoset(Ids subrelstr B) by
YELLOW_2:41;
    x1 is compact by A2,WAYBEL13:12;
    hence thesis by WAYBEL_8:def 1;
  end;
end;
