PROP. I. Substance is by nature prior to its modifications.
Proof.-This is clear from Deff. iii. and v.

PROP. II. Two substances, whose attributes 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 one
from the other, except the substances, or, in other words (see
Ax. iv.), their attributes and modifications. Q.E.D.

PROP. V. There cannot exist in the universe two or more
substances having the same nature or attribute.
Proof.-If several distinct substances be granted, they must