Q.E.D.
(6:5) This is shown still more readily by the absurdity of the
contradictory. (6) For, if substance be produced by an external cause,
the knowledge of it would depend on the knowledge of its cause ([A.iv] ),
and (by [D.iii] ) it would itself not be substance.
PROP. [VII] Existence belongs to the nature of substance.
Proof.- (7:1) 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.- (8:1) There can be only one substance with an identical attribute,
and existence follows from its nature ([vii] ); its nature, therefore,
involves existence, either as finite or infinite. (2) It does not exist
as finite, for (by [D.ii] ) it would then be limited by something else of
the same kind, which would also necessarily exist ([vii] ); and there
would be two substances with an identical attribute, which is absurd ([v] ).
(3) It therefore exists as infinite. Q.E.D.
Note [N.I]- (8:4) As finite existence involves a partial negation, and
infinite existence is the absolute affirmation of the given nature,
it follows (solely from [vii] ) that every substance is necessarily
infinite.
Note [N.II]- (8:5) No doubt it will be difficult for those who think about
things loosely, and have not been accustomed to know them by their primary
causes, to comprehend the demonstrations of [vii] : for such persons
make no distinction between the modifications of substances and the
substances themselves, and are ignorant of the manner in which things are
produced; hence they attribute to substances the beginning which they
observe in natural objects.
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