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

theorem Th8:
  for F being Function-yielding Function
  for f being Function st f in dom Frege F holds
    dom f = dom F & dom F = dom((Frege F).f)
proof
  let F be Function-yielding Function;
  let f be Function;
  assume f in dom Frege F ;
  then
A1: f in product doms F;
  then ex g being Function st g = f & dom g = dom(doms F) &
    for x being object st x in dom doms F holds g.x in (doms F).x
    by CARD_3:def 5;
  hence
A2: dom f = dom F by FUNCT_6:59;
  thus dom ((Frege F).f) = dom(F..f) by A1,PRALG_2:def 2
    .= dom F /\ dom f by PRALG_1:def 19
    .= dom F by A2;
end;
