
theorem Th32:
  for L being Semilattice, F being Filter of L, G being
GeneratorSet of F for f, g being sequence of  the carrier of L st rng f = G
& for n being Element of NAT holds g.n = "/\"({f.m where m is Element of NAT: m
  <= n},L) holds rng g is GeneratorSet of F
proof
  let L be Semilattice, F be Filter of L, G be GeneratorSet of F, f, g be
  sequence of  the carrier of L such that
A1: rng f = G and
A2: for n being Element of NAT holds g.n = "/\"({f.m where m is Element
  of NAT: m <= n},L);
A3: rng g is_coarser_than F
  proof
    let a be Element of L;
    assume a in rng g;
    then consider n being object such that
A4: n in dom g and
A5: g.n = a by FUNCT_1:def 3;
    reconsider n as Element of NAT by A4;
    reconsider Y = {f.m where m is Element of NAT: m <= n} as non empty finite
    Subset of L by Lm1;
A6: Y c= G
    proof
      let q be object;
      assume q in Y;
      then
A7:   ex m being Element of NAT st q = f.m & m <= n;
      dom f = NAT by FUNCT_2:def 1;
      hence thesis by A1,A7,FUNCT_1:def 3;
    end;
    G c= F by Lm4;
    then Y c= F by A6;
    then
A8: "/\"(Y,L) in F by WAYBEL_0:43;
    g.n = "/\"(Y,L) by A2;
    hence ex b being Element of L st b in F & b <= a by A5,A8;
  end;
  g.0 in rng g by FUNCT_2:4;
  hence thesis by A1,A2,A3,Th30,Th31;
end;
