
theorem Th2:
  for f,g being non-empty Function st product f = product g holds f = g
proof
  let f,g be non-empty Function;
  assume
A1: product f = product g;
  set h = the Element of product f;
A2: ex i being Function st h = i & dom i = dom g &
for x being object st x in
  dom g holds i.x in g.x by A1,CARD_3:def 5;
A3: ex i being Function st h = i & dom i = dom f &
    for x being object st x in dom f holds i.x in f.x by CARD_3:def 5;
  then for x be object st x in dom f holds f.x = g.x by A1,A2,Th1;
  hence thesis by A2,A3;
end;
