reserve x, X, Y for set;
reserve L for complete LATTICE,
  a for Element of L;

theorem Th47:
  for L being with_suprema Poset holds ex_inf_of {},InclPoset(Ids
  L) & "/\"({}, InclPoset(Ids L)) = [#]L
proof
  let L be with_suprema Poset;
  set P = InclPoset(Ids L);
  reconsider I = [#]L as Element of P by Th41;
A1: for b being Element of P st b is_<=_than {} holds I >= b
  proof
    let b be Element of P;
    reconsider b9= b as Ideal of L by Th41;
    assume {} is_>=_than b;
    b9 c= [#]L;
    hence thesis by YELLOW_1:3;
  end;
  I is_<=_than {};
  hence thesis by A1,YELLOW_0:31;
end;
