reserve x, y, i for object,
  L for up-complete Semilattice;

theorem Th14:
  for L being non empty RelStr for J being non empty set, K being
ManySortedSet of J for F being DoubleIndexedSet of K, L holds (x in rng Sups F
iff ex j being Element of J st x = Sup(F.j)) & (x in rng Infs F iff ex j being
  Element of J st x = Inf(F.j))
proof
  let L be non empty RelStr;
  let J be non empty set, K be ManySortedSet of J;
  let F be DoubleIndexedSet of K, L;
A1: dom F = J by PARTFUN1:def 2;
  thus x in rng Sups F iff ex j being Element of J st x = Sup(F.j)
  proof
    hereby
      assume x in rng Sups F;
      then consider j being object such that
A2:   j in dom F and
A3:   x = \\/(F.j, L) by Th13;
      reconsider j as Element of J by A2;
      take j;
      thus x = Sup(F.j) by A3;
    end;
    thus thesis by A1,Th13;
  end;
  hereby
    assume x in rng Infs F;
    then consider j being object such that
A4: j in dom F and
A5: x = //\(F.j, L) by Th13;
    reconsider j as Element of J by A4;
    take j;
    thus x = Inf(F.j) by A5;
  end;
  thus thesis by A1,Th13;
end;
