
theorem Th46: :: PROPOSITION 4.2. (1) iff (2)
  for L be continuous lower-bounded LATTICE for B be join-closed
  Subset of L holds B is CLbasis of L iff 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;
  thus B is CLbasis of L implies 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
A1: B is CLbasis of L;
    let x,y be Element of L such that
A2: not y <= x;
    thus ex b be Element of L st b in B & not b <= x & b << y
    proof
      assume
A3:   for b1 be Element of L holds not b1 in B or b1 <= x or not b1 << y;
A4:   waybelow y /\ B c= downarrow x
      proof
        let z be object;
        assume
A5:     z in waybelow y /\ B;
        then reconsider z1 = z as Element of L;
        z in waybelow y by A5,XBOOLE_0:def 4;
        then
A6:     z1 << y by WAYBEL_3:7;
        z in B by A5,XBOOLE_0:def 4;
        then z1 <= x by A3,A6;
        hence thesis by WAYBEL_0:17;
      end;
A7:   ex_sup_of downarrow x,L by YELLOW_0:17;
      ex_sup_of waybelow y /\ B,L by YELLOW_0:17;
      then sup (waybelow y /\ B) <= sup (downarrow x) by A7,A4,YELLOW_0:34;
      then y <= sup (downarrow x) by A1,Def7;
      hence contradiction by A2,WAYBEL_0:34;
    end;
  end;
  assume
A8: 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;
  now
    let x be Element of L;
A9: ex_sup_of waybelow x,L by YELLOW_0:17;
    ex_sup_of waybelow x /\ B,L by YELLOW_0:17;
    then
A10: sup (waybelow x /\ B) <= sup waybelow x by A9,XBOOLE_1:17,YELLOW_0:34;
A11: x <= sup (waybelow x /\ B)
    proof
      assume not x <= sup (waybelow x /\ B);
      then consider b be Element of L such that
A12:  b in B and
A13:  not b <= sup (waybelow x /\ B) and
A14:  b << x by A8;
      b in waybelow x by A14,WAYBEL_3:7;
      then b in waybelow x /\ B by A12,XBOOLE_0:def 4;
      hence contradiction by A13,YELLOW_2:22;
    end;
    x = sup waybelow x by WAYBEL_3:def 5;
    hence x = sup (waybelow x /\ B) by A10,A11,YELLOW_0:def 3;
  end;
  hence thesis by Def7;
end;
