theorem Th8:
  x in product G iff for i holds x.i in G.i
proof
  x is Function of Seg d,REAL by Def3;
  then
A1: dom x = Seg d by FUNCT_2:def 1;
A2: dom G = Seg d by FUNCT_2:def 1;
  hence x in product G implies for i holds x.i in G.i by CARD_3:9;
  assume for i holds x.i in G.i;
  then for i being object st i in Seg d holds x.i in G.i;
  hence thesis by A1,A2,CARD_3:9;
end;
