are different, have nothing in common.
Proof.—Also evident from Def. iii. For each must exist in itself, and be conceived through itself; in other words, the conception of one does not imply the conception of the other.
PROP. III. Things which have nothing in common cannot be one the cause of the other.
Proof.—If they have nothing in common, it follows that one cannot be apprehended by means of the other (Ax. v.), and, therefore, one cannot be the cause of the other (Ax. iv.). Q.E.D.
PROP. IV. Two or more distinct things are distinguished one from the other, either by the difference of the attributes of the substances, or by the difference of their modifications.
Proof.—Everything which exists, exists either in itself or in something else (Ax. i.),—that is (by Deff. iii. and v.), nothing is granted in addition to the understanding, except substance and its modifications. Nothing is, therefore, given besides the understanding, by which several things may be distinguished