
theorem Th31:
  for L being Semilattice, A being Subset of L for f, g being
  sequence of  the carrier of L st rng f = A & for n being Element of NAT
  holds g.n = "/\"({f.m where m is Element of NAT: m <= n},L) holds A
  is_coarser_than rng g
proof
  let L be Semilattice, A be Subset of L, f, g be sequence of  the carrier
  of L such that
A1: rng f = A and
A2: for n being Element of NAT holds g.n = "/\"({f.m where m is Element
  of NAT: m <= n},L);
  let a be Element of L;
  assume a in A;
  then consider n being object such that
A3: n in dom f and
A4: f.n = a by A1,FUNCT_1:def 3;
  reconsider n as Element of NAT by A3;
  reconsider T = {f.m where m is Element of NAT: m <= n} as non empty finite
  Subset of L by Lm1;
  take b = "/\"(T,L);
  dom g = NAT & g.n = b by A2,FUNCT_2:def 1;
  hence b in rng g by FUNCT_1:def 3;
  f.n in T;
  then
A5: {f.n} c= T by ZFMISC_1:31;
  ex_inf_of {f.n},L & ex_inf_of T,L by YELLOW_0:55;
  then b <= "/\"({f.n},L) by A5,YELLOW_0:35;
  hence b <= a by A4,YELLOW_0:39;
end;
