
theorem Th5:
  for T be non empty TopSpace for L1 be continuous lower-bounded
  LATTICE st InclPoset the topology of T = L1 for B1 be Basis of T for B2 be
  Subset of L1 st B1 = B2 holds finsups B2 is with_bottom CLbasis of L1
proof
  let T be non empty TopSpace;
  let L1 be continuous lower-bounded LATTICE;
  assume
A1: InclPoset the topology of T = L1;
  let B1 be Basis of T;
  let B2 be Subset of L1;
  assume
A2: B1 = B2;
A3: for x,y be Element of L1 st not y <= x ex b be Element of L1 st b in
  finsups B2 & not b <= x & b <= y
  proof
    let x,y be Element of L1;
    y in the carrier of L1;
    then
A4: y in the topology of T by A1,YELLOW_1:1;
    then reconsider y1 = y as Subset of T;
    assume not y <= x;
    then not y c= x by A1,YELLOW_1:3;
    then consider v be object such that
A5: v in y and
A6: not v in x;
    v in y1 by A5;
    then reconsider v as Point of T;
    y1 is open by A4,PRE_TOPC:def 2;
    then consider b be Subset of T such that
A7: b in B1 and
A8: v in b and
A9: b c= y1 by A5,YELLOW_9:31;
    reconsider b as Element of L1 by A2,A7;
    for z be object st z in {b} holds z in B2 by A2,A7,TARSKI:def 1;
    then
A10: {b} is finite Subset of B2 by TARSKI:def 3;
    take b;
    ex_sup_of {b},L1 & b = "\/"({b},L1) by YELLOW_0:38,39;
    then
    b in { "\/"(Y,L1) where Y is finite Subset of B2: ex_sup_of Y,L1 } by A10;
    hence b in finsups B2 by WAYBEL_0:def 27;
    not b c= x by A6,A8;
    hence not b <= x by A1,YELLOW_1:3;
    thus thesis by A1,A9,YELLOW_1:3;
  end;
  now
    let x,y be Element of L1;
    assume that
A11: x in finsups B2 and
A12: y in finsups B2;
    y in { "\/"(Y,L1) where Y is finite Subset of B2: ex_sup_of Y,L1 } by A12,
WAYBEL_0:def 27;
    then consider Y2 be finite Subset of B2 such that
A13: y = "\/"(Y2,L1) and
A14: ex_sup_of Y2,L1;
    x in { "\/"(Y,L1) where Y is finite Subset of B2: ex_sup_of Y,L1 } by A11,
WAYBEL_0:def 27;
    then consider Y1 be finite Subset of B2 such that
A15: x = "\/"(Y1,L1) and
A16: ex_sup_of Y1,L1;
    ex_sup_of (Y1 \/ Y2),L1 & "\/"(Y1 \/ Y2, L1) = "\/"(Y1, L1) "\/" "\/"
    (Y2, L1) by A16,A14,YELLOW_2:3;
    then
    x "\/" y in { "\/" (Y,L1) where Y is finite Subset of B2: ex_sup_of Y
    ,L1 } by A15,A13;
    then x "\/" y in finsups B2 by WAYBEL_0:def 27;
    hence sup {x,y} in finsups B2 by YELLOW_0:41;
  end;
  then
A17: finsups B2 is join-closed by WAYBEL23:18;
  {} c= B2 & ex_sup_of {},L1 by YELLOW_0:42;
  then
  "\/"({},L1) in {"\/"(Y,L1) where Y is finite Subset of B2: ex_sup_of Y, L1};
  then Bottom L1 in finsups B2 by WAYBEL_0:def 27;
  hence thesis by A17,A3,WAYBEL23:49,def 8;
end;
