What is discussed on this page has to do with other subsidiaries. For example:
[IF ONE CARRIES OUT] A LIVING PERSON IN A BED, HE IS NOT CULPABLE EVEN IN RESPECT OF THE BED (BECAUSE THE BED IS SUBSIDIARY TO HIM)
That subsidiary idea carries to animals as well. However there is a discussion about the difference between a live animal, which could carry itself, and a dead one which “stiffens.”
An analogy is given in respect to a living horse. One is not allowed to sell cattle to a Gentile because it may be used for plowing on Shabbat. A horse, though, may be sold – because at that time they were not used as draught animals, only for riding. Similarly – a horse carries itself!