
theorem :: Under 4.1 (ii)
  for L be continuous lower-bounded sup-Semilattice st the carrier of
  CompactSublatt L is CLbasis of L holds L is algebraic
proof
  let L be continuous lower-bounded sup-Semilattice;
  reconsider C = the carrier of CompactSublatt L as Subset of L by Th43;
  assume
A1: the carrier of CompactSublatt L is CLbasis of L;
  now
    let x be Element of L;
    x = sup (waybelow x /\ C) by A1,Def7;
    hence x = sup compactbelow x by Th1;
  end;
  then
A2: L is satisfying_axiom_K by WAYBEL_8:def 3;
  for x being Element of L holds compactbelow x is non empty directed;
  hence thesis by A2,WAYBEL_8:def 4;
end;
