A = ABC . C.
Substituting again for ABC its equivalent A, we obtain
A = AC,
the desired result.
By a similar process of reasoning it may be shown that we can always drop out any term appearing in one member of a proposition, provided that we substitute for it the whole of the other member. This process was described in my first logical Essay,[62] as Intrinsic Elimination, but it might perhaps be better entitled the Ellipsis of Terms. It enables us to get rid of needless terms by strict substitutive reasoning.
Inference of a Simple from Two Partial Identities.
Two terms may be connected together by two partial identities in yet another manner, and a case of inference then arises which is of the highest importance. In the two premises
A = AB (1)
B = AB (2)
the second member of each is the same; so that we can by obvious substitution obtain
A = B.