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

theorem Th9:
  for F being Function-yielding Function
  for f being Function st f in dom Frege F
  for i being set st i in dom F holds f.i in dom(F.i) &
  ((Frege F).f).i = (F.i).(f.i) & (F.i).(f.i) in rng((Frege F).f)
proof
  let F be Function-yielding Function;
  let f be Function such that
A1: f in dom Frege F;
A2: f in product doms F by A1;
  set G = (Frege F).f;
  let i be set such that
A3: i in dom F;
  i in dom f by Th8,A1,A3; then
  i in dom F /\ dom f by A3,XBOOLE_0:def 4; then
a3: i in dom (F..f) by PRALG_1:def 19;
  i in dom doms F by A3,FUNCT_6:59;
  then f.i in (doms F).i by A2,CARD_3:9;
  hence f.i in dom(F.i) by A3,FUNCT_6:22;
  G = F..f by A2,PRALG_2:def 2;
  hence
A4: G.i = (F.i).(f.i) by a3,PRALG_1:def 19;
  dom G = dom F by A1,Th8;
  hence thesis by A3,A4,FUNCT_1:def 3;
end;
