
theorem Th21:
  for L1 be continuous lower-bounded sup-Semilattice for T be
Lawson correct TopAugmentation of L1 holds for B1 be with_bottom CLbasis of L1
  holds { Way_Up(a,A) where a is Element of L1, A is finite Subset of L1 : a in
  B1 & A c= B1 } is Basis of T
proof
  let L1 be continuous lower-bounded sup-Semilattice;
  let T be Lawson correct TopAugmentation of L1;
  let B1 be with_bottom CLbasis of L1;
A1: the RelStr of L1 = the RelStr of T by YELLOW_9:def 4;
  { Way_Up(a,A) where a is Element of L1, A is finite Subset of L1 : a in
  B1 & A c= B1 } c= bool the carrier of T
  proof
    let z be object;
    assume z in { Way_Up(a,A) where a is Element of L1, A is finite Subset of
L1: a in B1 & A c= B1 };
    then
    ex a be Element of L1, A be finite Subset of L1 st z = Way_Up(a,A) & a
    in B1 & A c= B1;
    hence thesis by A1;
  end;
  then reconsider
  WU = { Way_Up(a,A) where a is Element of L1, A is finite Subset
  of L1 : a in B1 & A c= B1 } as Subset-Family of T;
  reconsider WU as Subset-Family of T;
A2: now
    reconsider BL = { W \ uparrow F where W,F is Subset of T : W in sigma T &
    F is finite} as Basis of T by WAYBEL19:32;
    set S = the Scott TopAugmentation of T;
    let A be Subset of T;
    assume
A3: A is open;
    let pT be Point of T;
    assume pT in A;
    then consider a be Subset of T such that
A4: a in BL and
A5: pT in a and
A6: a c= A by A3,YELLOW_9:31;
    consider W,FT be Subset of T such that
A7: a = W \ uparrow FT and
A8: W in sigma T and
A9: FT is finite by A4;
A10: the RelStr of S = the RelStr of T by YELLOW_9:def 4;
    then reconsider pS = pT as Element of S;
    reconsider W1 = W as Subset of S by A10;
    sigma S = sigma T by A10,YELLOW_9:52;
    then
A11: W = union { wayabove x where x is Element of S : x in W1 } by A8,
WAYBEL14:33;
    reconsider pL = pS as Element of L1 by A1;
    defpred P[object,object] means
ex b1,y1 be Element of L1 st y1 = $1 & b1 = $2 &
    b1 in B1 & not b1 <= pL & b1 << y1;
A12: Bottom L1 in B1 by WAYBEL23:def 8;
    pT in W by A5,A7,XBOOLE_0:def 5;
    then consider k be set such that
A13: pT in k and
A14: k in { wayabove x where x is Element of S : x in W1 } by A11,TARSKI:def 4;
    consider xS be Element of S such that
A15: k = wayabove xS and
A16: xS in W1 by A14;
    reconsider xL = xS as Element of L1 by A1,A10;
    xS << pS by A13,A15,WAYBEL_3:8;
    then xL << pL by A1,A10,WAYBEL_8:8;
    then consider bL be Element of L1 such that
A17: bL in B1 and
A18: xL <= bL and
A19: bL << pL by A12,WAYBEL23:47;
    reconsider FL = FT as Subset of L1 by A1;
A20: uparrow FT = uparrow FL by A1,WAYBEL_0:13;
A21: not pT in uparrow FT by A5,A7,XBOOLE_0:def 5;
A22: for y be object st y in FL ex b be object st P[y,b]
    proof
      let y be object;
      assume
A23:  y in FL;
      then reconsider y1 = y as Element of L1;
      not y1 <= pL by A21,A20,A23,WAYBEL_0:def 16;
      then consider b1 be Element of L1 such that
A24:  b1 in B1 & not b1 <= pL & b1 << y1 by WAYBEL23:46;
      reconsider b = b1 as set;
      take b,b1,y1;
      thus thesis by A24;
    end;
    consider f be Function such that
A25: dom f = FL and
A26: for y be object st y in FL holds P[y,f.y] from CLASSES1:sch 1(A22);
    rng f c= the carrier of L1
    proof
      let z be object;
      assume z in rng f;
      then consider v be object such that
A27:  v in dom f and
A28:  z = f.v by FUNCT_1:def 3;
      ex b1,y1 be Element of L1 st y1 = v & b1 = f.v & b1 in B1 &( not b1
      <= pL)& b1 << y1 by A25,A26,A27;
      hence thesis by A28;
    end;
    then reconsider FFL = rng f as Subset of L1;
A29: FFL c= B1
    proof
      let z be object;
      assume z in FFL;
      then consider v be object such that
A30:  v in dom f and
A31:  z = f.v by FUNCT_1:def 3;
      ex b1,y1 be Element of L1 st y1 = v & b1 = f.v & b1 in B1 &( not b1
      <= pL)& b1 << y1 by A25,A26,A30;
      hence thesis by A31;
    end;
A32: uparrow FL c= uparrow FFL
    proof
      let z be object;
      assume
A33:  z in uparrow FL;
      then reconsider z1 = z as Element of L1;
      consider v1 be Element of L1 such that
A34:  v1 <= z1 and
A35:  v1 in FL by A33,WAYBEL_0:def 16;
      consider b1,y1 be Element of L1 such that
A36:  y1 = v1 and
A37:  b1 = f.v1 and
      b1 in B1 and
      not b1 <= pL and
A38:  b1 << y1 by A26,A35;
      b1 << z1 by A34,A36,A38,WAYBEL_3:2;
      then
A39:  b1 <= z1 by WAYBEL_3:1;
      b1 in FFL by A25,A35,A37,FUNCT_1:def 3;
      hence thesis by A39,WAYBEL_0:def 16;
    end;
    reconsider cT = wayabove bL \ uparrow FFL as Subset of T by A1;
    take cT;
    cT = Way_Up(bL,FFL) & FFL is finite by A9,A25,FINSET_1:8;
    hence cT in WU by A17,A29;
    wayabove bL c= wayabove xL by A18,WAYBEL_3:12;
    then
A40: wayabove bL \ uparrow FFL c= wayabove xL \ uparrow FL by A32,XBOOLE_1:35;
    for z be Element of L1 holds not z in FFL or not z <= pL
    proof
      let z be Element of L1;
      assume z in FFL;
      then consider v be object such that
A41:  v in dom f and
A42:  z = f.v by FUNCT_1:def 3;
      ex b1,y1 be Element of L1 st y1 = v & b1 = f.v & b1 in B1 &( not b1
      <= pL)& b1 << y1 by A25,A26,A41;
      hence thesis by A42;
    end;
    then for z be Element of L1 holds not z <= pL or not z in FFL;
    then
A43: not pL in uparrow FFL by WAYBEL_0:def 16;
    pL in wayabove bL by A19,WAYBEL_3:8;
    hence pT in cT by A43,XBOOLE_0:def 5;
    wayabove xL c= W
    proof
      let z be object;
      wayabove xL = wayabove xS by A1,A10,YELLOW12:13;
      then
A44:  wayabove xL in { wayabove x where x is Element of S : x in W1 } by A16;
      assume z in wayabove xL;
      hence thesis by A11,A44,TARSKI:def 4;
    end;
    then wayabove xL \ uparrow FL c= a by A7,A20,XBOOLE_1:33;
    then wayabove bL \ uparrow FFL c= a by A40;
    hence cT c= A by A6;
  end;
  WU c= the topology of T
  proof
    let z be object;
    assume z in WU;
    then consider a be Element of L1, A be finite Subset of L1 such that
A45: z = Way_Up(a,A) and
    a in B1 and
    A c= B1;
    reconsider A1 = A as finite Subset of T by A1;
    reconsider a1 = a as Element of T by A1;
    wayabove a1 is open & (uparrow A1)` is open by Th20,WAYBEL19:40;
    then
A46: wayabove a1 /\ (uparrow A1)` is open by TOPS_1:11;
    z = wayabove a1 \ uparrow A by A1,A45,YELLOW12:13
      .= wayabove a1 \ uparrow A1 by A1,WAYBEL_0:13
      .= wayabove a1 /\ (uparrow A1)` by SUBSET_1:13;
    hence thesis by A46,PRE_TOPC:def 2;
  end;
  hence thesis by A2,YELLOW_9:32;
end;
