
theorem Th48: :: PROPOSITION 4.2. (1) iff (4)
  for L be continuous lower-bounded LATTICE for B be join-closed
  Subset of L st Bottom L in B holds B is CLbasis of L iff (the carrier of
  CompactSublatt L c= B & for x,y be Element of L st not y <= x ex b be Element
  of L st b in B & not b <= x & b <= y)
proof
  let L be continuous lower-bounded LATTICE;
  let B be join-closed Subset of L;
  assume
A1: Bottom L in B;
  thus B is CLbasis of L implies (the carrier of CompactSublatt L c= B & for x
,y be Element of L st not y <= x ex b be Element of L st b in B & not b <= x &
  b <= y)
  proof
    assume B is CLbasis of L;
    then for x,y be Element of L st x << y ex b be Element of L st b in B & x
    <= b & b << y by A1,Th47;
    hence thesis by A1,Lm2;
  end;
  assume that
  the carrier of CompactSublatt L c= B and
A2: for x,y be Element of L st not y <= x ex b be Element of L st b in B
  & not b <= x & b <= y;
  for x,y be Element of L st not y <= x ex b be Element of L st b in B &
  not b <= x & b << y by A2,Lm3;
  hence thesis by Th46;
end;
