theorem Th10:
  product {} = {{}}
proof
  thus product {} c= {{}}
  proof
    let x be object;
    assume x in product {};
    then ex g st
    x = g & dom g = dom {} &
     for y being object st y in dom {} holds g.y in {} .y by Def5;
    then x = {};
    hence thesis by TARSKI:def 1;
  end;
  let x be object;
  assume x in {{}};
  then
A1: x = {} by TARSKI:def 1;
  for y being object st y in dom {} holds {} .y in {} .y;
  hence thesis by A1,Th9;
end;
