reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;

theorem Th9:
  g in product f iff dom g = dom f &
   for x being object st x in dom f holds g.x in f.x
proof
  thus g in product f implies
  dom g = dom f & for x being object st x in dom f holds g.x in f.x
  proof
    assume g in product f;
    then ex h being Function st
    g = h & dom h = dom f &
     for x being object st x in dom f holds h.x in f.x by Def5;
    hence thesis;
  end;
  thus thesis by Def5;
end;
