
theorem Th44: :: Under 4.1 (i)
  for L be algebraic lower-bounded LATTICE holds the carrier of
  CompactSublatt L is with_bottom CLbasis of L
proof
  let L be algebraic lower-bounded LATTICE;
  reconsider C = the carrier of CompactSublatt L as join-closed Subset of L by
Th43;
  now
    let x be Element of L;
    x = sup compactbelow x by WAYBEL_8:def 3;
    hence x = sup (waybelow x /\ C) by Th1;
  end;
  then reconsider C as CLbasis of L by Def7;
  Bottom L in C by WAYBEL_8:3;
  hence thesis by Def8;
end;
