er 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
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.

PROP. VI. One substance cannot be produced by another substance.

Proof.--It is impossible that there should be in the universe
two substances with an identical attribute, i.e. which have
anything common to them both (Prop. ii.), and, therefore (Prop.
iii.), one cannot be the cause of the other, neither can one be
produced by the other. Q.E.D.

Corollary.--Hence it follows that a substance cannot be
produced by anything external to itself. For in the universe
nothing is granted, save substances and their modifications (as
appears from Ax. i. and Deff. iii. and v.). Now (by the last
Prop.) substance cannot be produced by another substance,
therefore it cannot be produced by anything external to itself.
Q.E.D. This is shown still more readily by the absurdity of the
contradictory. For, if substance be produced by an external
cause, the knowledge of it would depend on the knowledge of its
cause (Ax. iv.), and (by Def. iii.) it would itself not be
substance.

PROP. VII. Existence belongs to the nature of substances.

Proof.--Substance cannot be produced by anything external
(Corollary, Prop vi.), it must, therefore, be its own cause--that
is, its essence necessarily involves existence, or existence
belongs to its nature.

PROP. VIII. Every substance is necessarily infinite.

Proof.--There can only be one substance with an identical
attribute, and existence follows from its nature (Prop. vii.);
its nature, therefore, involves existence, either as finite or
infinite. It does not exist as finite, for (by Def. ii.)