reserve x,y,y1,y2,z,a,b for object, X,Y,Z,V1,V2 for set,
  f,g,h,h9,f1,f2 for Function,
  i for Nat,
  P for Permutation of X,
  D,D1,D2,D3 for non empty set,
  d1 for Element of D1,
  d2 for Element of D2,
  d3 for Element of D3;

theorem
  for f being Function-yielding Function
   st x in dom f & g = f.x & h in product doms f & h9 = (Frege f).h
  holds h.x in dom g & h9.x = g.(h.x) & h9 in product rngs f
proof let f be Function-yielding Function;
  assume that
A1: x in dom f & g = f.x and
A2: h in product doms f and
A3: h9 = (Frege f).h;
A4: x in dom f by A1;
A5: dom doms f = dom f by Def1;
  ( ex f2 st h = f2 & dom f2 = dom doms f &
for x being object st x in dom doms f holds
  f2.x in (doms f).x)& (doms f).x = dom g by A1,A2,Th18,CARD_3:def 5;
  hence
A6: h.x in dom g by A5,A4;
  ex f1 st (Frege f).h = f1 & dom f1 = dom f & for x st x in
  dom f1 holds f1.x = (uncurry f).(x,h.x) by A2,Def6;
  hence h9.x = (uncurry f).(x,h.x) by A3,A4
    .= g.(h.x) by A1,A6,FUNCT_5:38;
A7: rng Frege f c= product rngs f by Lm1;
  dom Frege f = product doms f by Def6;
  then h9 in rng Frege f by A2,A3,FUNCT_1:def 3;
  hence thesis by A7;
end;
