
theorem Th4:
  for T be non empty TopSpace for L1 be continuous sup-Semilattice
  st InclPoset the topology of T = L1 for B1 be with_bottom CLbasis of L1 holds
  B1 is Basis of T
proof
  let T be non empty TopSpace;
  let L1 be continuous sup-Semilattice;
  assume
A1: InclPoset the topology of T = L1;
  let B1 be with_bottom CLbasis of L1;
  B1 c= the carrier of L1;
  then B1 c= the topology of T by A1,YELLOW_1:1;
  then reconsider B2 = B1 as Subset-Family of T by XBOOLE_1:1;
A2: for A be Subset of T st A is open for p be Point of T st p in A ex a be
  Subset of T st a in B2 & p in a & a c= A
  proof
    let A be Subset of T;
    assume A is open;
    then A in the topology of T by PRE_TOPC:def 2;
    then reconsider A1 = A as Element of L1 by A1,YELLOW_1:1;
    let p be Point of T;
    assume
A3: p in A;
    A1 = sup (waybelow A1 /\ B1) by WAYBEL23:def 7
      .= union (waybelow A1 /\ B1) by A1,YELLOW_1:22;
    then consider a be set such that
A4: p in a and
A5: a in waybelow A1 /\ B1 by A3,TARSKI:def 4;
    a in the carrier of L1 by A5;
    then a in the topology of T by A1,YELLOW_1:1;
    then reconsider a as Subset of T;
    take a;
    thus a in B2 by A5,XBOOLE_0:def 4;
    thus p in a by A4;
    reconsider a1 = a as Element of L1 by A5;
    a1 in waybelow A1 by A5,XBOOLE_0:def 4;
    then a1 << A1 by WAYBEL_3:7;
    then a1 <= A1 by WAYBEL_3:1;
    hence thesis by A1,YELLOW_1:3;
  end;
  B2 c= the topology of T
  proof
    let x be object;
    assume x in B2;
    then reconsider x1 = x as Element of L1;
    x1 in the carrier of L1;
    hence thesis by A1,YELLOW_1:1;
  end;
  hence thesis by A2,YELLOW_9:32;
end;
