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

theorem Th48:
  for L being lower-bounded sup-Semilattice holds InclPoset(Ids L) is complete
proof
  let L be lower-bounded sup-Semilattice;
  set P = InclPoset(Ids L);
  for A being Subset of P holds ex_inf_of A,InclPoset(Ids L)
  proof
    let A be Subset of P;
    per cases;
    suppose
      A = {};
      hence thesis by Th47;
    end;
    suppose
      A <> {};
      hence thesis by Th46;
    end;
  end;
  hence thesis by Th28;
end;
