theorem
  h in sproduct(f+*g) implies
  ex f9,g9 being Function st f9 in sproduct f & g9 in sproduct g & h = f9+*g9
proof
  assume
A1: h in sproduct(f+*g);
  take h|(dom f \ dom g), h|dom g;
A2: h|(dom f \ dom g) in sproduct (f +* g)|(dom f \ dom g) by A1,Th66;
  sproduct (f +* g)|(dom f \ dom g) c= sproduct f by Th56,FUNCT_4:24;
  hence h|(dom f \ dom g) in sproduct f by A2;
  (f +* g)|dom g = g;
  hence h|dom g in sproduct g by A1,Th66;
  dom h c= dom(f+*g) by A1,Th49;
  then dom h c= dom f \/ dom g by FUNCT_4:def 1;
  then dom h c= (dom f \ dom g) \/ dom g by XBOOLE_1:39;
  hence thesis by FUNCT_4:70;
end;
