VII. If a thing can be conceived as non-existing, its essence
does not involve existence.
PROPOSITIONS. I. Substance is by nature prior to its
modifications.
>>>>>Proof--This is clear from Deff. iii. and v.
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.
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.
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.
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
be distinguished one from the other, either by the difference of
their attributes, or by the difference of their modifications
(Prop. iv.). If only by the difference of their attributes, it
will be granted that there cannot be more than one with an
identical attribute. If by the difference of their
modifications--as substance is naturally prior to its
modifications (Prop. i.)--it follows that setting the
modifications aside, and considering substance in itself, that is
truly, (Deff. iii and vi.), there cannot be conceived one
substance different from another--that is (by Prop. iv.), there
cannot be granted several substances, but one substance only.
Q.E.D.