
theorem :: THEOREM 4.7 (5)
  for L1 be continuous lower-bounded sup-Semilattice holds CLweight L1
  c= CLweight InclPoset sigma L1
proof
  let L1 be continuous lower-bounded sup-Semilattice;
  set S = the Scott TopAugmentation of L1;
A1: the RelStr of S = the RelStr of L1 by YELLOW_9:def 4;
A2: InclPoset the topology of S = InclPoset sigma L1 by YELLOW_9:51;
A3: CLweight L1 = weight S by Th24;
  now
    per cases;
    suppose
      L1 is infinite;
      then S is infinite by A1;
      hence thesis by A3,A2,Th6;
    end;
    suppose
A4:   L1 is finite;
A5:   card the carrier of S c= card the carrier of InclPoset sigma L1 by A2,Th7
;
A6:   S is finite by A1,A4;
      then weight S = card the carrier of S by Th8;
      hence thesis by A3,A2,A6,A5,Th9;
    end;
  end;
  hence thesis;
end;
