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

theorem Th10:
  for J, D being set, K being ManySortedSet of J
  for F being DoubleIndexedSet of K, D
  for f being Function st f in dom(Frege F) holds
  (Frege F).f is Function of J, D
proof
  let J, D be set, K be ManySortedSet of J;
  let F be DoubleIndexedSet of K, D;
  let f be Function such that
A1: f in dom(Frege F);
  set G = (Frege F).f;
A2: dom G = dom F by A1,Th8;
A3: dom F = J by PARTFUN1:def 2;
  rng G c= D
  proof
    let y be object;
    assume y in rng G;
    then consider x being object such that
A4: x in dom G and
A5: y = G.x by FUNCT_1:def 3;
    F.x is Function of K.x, D by A2,A4,Th6;
    then
A6: rng(F.x) c= D by RELAT_1:def 19;
    G.x = (F.x).(f.x) & f.x in dom(F.x) by A1,A2,A4,Th9;
    then y in rng(F.x) by A5,FUNCT_1:def 3;
    hence thesis by A6;
  end;
  hence thesis by A3,A2,FUNCT_2:2;
end;
