
theorem Th1:
  for f,g being Function st product f = product g holds f is
  non-empty implies g is non-empty
proof
  let f,g be Function;
  assume
A1: product f = product g;
  now
    assume that
A2: f is non-empty and
A3: not g is non-empty;
    ex n being object st n in dom g & g.n is empty by A3,FUNCT_1:def 9;
    then {} in rng g by FUNCT_1:def 3;
    then product g = {} by CARD_3:26;
    then {} in rng f by A1,CARD_3:26;
    then ex n being object st n in dom f & f.n = {} by FUNCT_1:def 3;
    hence contradiction by A2,FUNCT_1:def 9;
  end;
  hence thesis;
end;
