
theorem Th23:
  for L1 be continuous lower-bounded sup-Semilattice for T be
Scott TopAugmentation of L1 for B1 be Basis of T st B1 is infinite holds { inf
  u where u is Subset of T : u in B1 } is infinite
proof
  let L1 be continuous lower-bounded sup-Semilattice;
  let T be Scott TopAugmentation of L1;
  let B1 be Basis of T;
  reconsider B2 = { inf u where u is Subset of T : u in B1 } as set;
  defpred P[object,object] means
   ex y be Element of T st $1 = y & $2 = wayabove y;
  reconsider B3 = { wayabove inf u where u is Subset of T : u in B1 } as Basis
  of T by Th22;
  assume that
A1: B1 is infinite and
A2: { inf u where u is Subset of T : u in B1 } is finite;
A3: for x be object st x in B2 ex y be object st y in B3 & P[x,y]
  proof
    let x be object;
    assume x in B2;
    then
A4: ex u1 be Subset of T st x = inf u1 & u1 in B1;
    then reconsider z = x as Element of T;
    take y = wayabove z;
    thus y in B3 by A4;
    take z;
    thus thesis;
  end;
  consider f be Function such that
A5: dom f = B2 and
A6: rng f c= B3 and
A7: for x be object st x in B2 holds P[x,f.x] from FUNCT_1:sch 6(A3);
  B3 c= rng f
  proof
    let z be object;
    assume z in B3;
    then consider u1 be Subset of T such that
A8: z = wayabove inf u1 and
A9: u1 in B1;
    inf u1 in B2 by A9;
    then
A10: ex y be Element of T st inf u1 = y & f.(inf u1) = wayabove y by A7;
    inf u1 in B2 by A9;
    hence thesis by A5,A8,A10,FUNCT_1:def 3;
  end;
  then rng f = B3 by A6,XBOOLE_0:def 10;
  then B3 is finite by A2,A5,FINSET_1:8;
  then T is finite by YELLOW15:30;
  hence contradiction by A1;
end;
