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 g in product doms f & x in dom g
 holds (Frege f)..(g,x) = f..(x,g.x)
proof let f be Function-yielding Function;
  assume that
A1: g in product doms f and
A2: x in dom g;
A3: dom g = dom doms f by A1,CARD_3:9;
A4: dom doms f = dom f by Def1;
  consider h such that
A5: (Frege f).g = h and
A6: dom h = dom f and
A7: for x st x in dom h holds h.x = (uncurry f).(x,g.x) by A1,Def6;
  dom Frege f = product doms f by Def6;
  hence (Frege f)..(g,x) = h.x by A1,A2,A5,A6,A3,A4,FUNCT_5:38
    .= f..(x,g.x) by A2,A6,A7,A3,A4;
end;
