It is to be seen whether Lux has registered their trademark in multiple classes. Or may be Lux Cozy would have registered their trademark under a different class in which Lux did not have registration.
Section 28 would throw some light in case both the trademarks are registered.
Rights conferred by Registration:
28 (1) Subject to the other provisions of this Act, the registration of a trade mark shall, if valid, give to the registered proprietor of the trade mark the exclusive right to the use of the trade mark in relation to the goods or services in respect of which the trade mark is registered and to obtain relief in respect of infringement of trade mark in the manner provided by this Act.
(2) The exclusive right to the use of a trade mark given under sub-section (1) shall be subject to any conditions and limitations to which the registration is subject.
(3) Where two or more persons are registered proprietors of trademarks, which are identical with or nearly resemble each other, the exclusive right to the use of any of those trade marks shall not (except so far as their respective rights are subject to any conditions or limitations entered on the register) be deemed to have been acquired by any one of those persons as against any other of those persons merely by registration of the trade marks but each of those persons has otherwise the same rights as against other persons (not being registered users using by way of permitted use) as he would have if he were the sole registered proprietor.