
theorem Th47: :: PROPOSITION 4.2. (1) iff (3)
  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 for x,y be Element of
  L st x << y ex b be Element of L st b in B & x <= b & 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 for x,y be Element of L st x << y ex b be
  Element of L st b in B & x <= b & b << y
  proof
    assume
A2: B is CLbasis of L;
    let x,y be Element of L;
    assume
A3: x << y;
    Bottom L << y by WAYBEL_3:4;
    then
A4: Bottom L in waybelow y by WAYBEL_3:7;
    waybelow y /\ B is join-closed by Th33;
    then reconsider
    D = waybelow y /\ B as non empty directed Subset of L by A1,A4,
XBOOLE_0:def 4;
    y = sup D by A2,Def7;
    then consider b be Element of L such that
A5: b in D and
A6: x <= b by A3,WAYBEL_3:def 1;
    take b;
    b in waybelow y by A5,XBOOLE_0:def 4;
    hence thesis by A5,A6,WAYBEL_3:7,XBOOLE_0:def 4;
  end;
  assume
A7: for x,y be Element of L st x << y ex b be Element of L st b in B &
  x <= b & b << y;
  now
    let x be Element of L;
A8: x <= sup (waybelow x /\ B)
    proof
A9:   for x being Element of L holds waybelow x is non empty directed;
      assume not x <= sup (waybelow x /\ B);
      then consider u be Element of L such that
A10:  u << x and
A11:  not u <= sup (waybelow x /\ B) by A9,WAYBEL_3:24;
      consider b be Element of L such that
A12:  b in B and
A13:  u <= b and
A14:  b << x by A7,A10;
      b in waybelow x by A14,WAYBEL_3:7;
      then
A15:  b in waybelow x /\ B by A12,XBOOLE_0:def 4;
A16:  sup (waybelow x /\ B) is_>=_than waybelow x /\ B by YELLOW_0:32;
      not b <= sup (waybelow x /\ B) by A11,A13,YELLOW_0:def 2;
      hence contradiction by A15,A16;
    end;
    waybelow x /\ B c= waybelow x by XBOOLE_1:17;
    then sup (waybelow x /\ B) <= sup waybelow x by WAYBEL_7:1;
    then sup (waybelow x /\ B) <= x by WAYBEL_3:def 5;
    hence x = sup (waybelow x /\ B) by A8,YELLOW_0:def 3;
  end;
  hence thesis by Def7;
end;
