V. It is impossible that there should be two or more substances of the same nature, or of the same attributes. Dem. If there are many different substances they must be distinguished by the diversity of their attributes or of their modes (per prop. 4.) If only by the diversity of their attributes, it is thereby conceded that there is, nevertheless, only one substance of the same attribute; but if their diversity of modes, then, substance being prior in order of time to its modes, it must be considered independent of them--that is (per def. three and six,) cannot be conceived as distinguished from another--that is (per prop, four,) there cannot be many substances, but only one substance.--Q. E. D. VI. One substance cannot be created by another substance. Dem. There cannot be two substances with the same attributes (per prop, five)--that is (per prop. two,) that hare anything in common with each other; and, therefore (per prop, three,) one cannot be the cause of the other. Corollary 1. Hence it follows that substance cannot be created by anything else. For there is nothing in nature except substance and its modes (per axiom one, and def. three and five.) Now, this substance, not being produced by another, is self-caused. Corollary 2. This proposition is more easily to be demonstrated by the absurdity of its contradiction; for if substance can be produced by anything else, the conception of it would depend on the conception of the cause (per axiom four,) and hence (per def. three,) it would not be substance. VII. It pertains to the nature of substance to exist. Dem. Substance cannot be produced by anything else (per coroli. prop, six,) and is therefore the cause of itself--that is (per def. one,) its essence necessarily involves existence; or it pertains to the nature of substance to exist.--Q. E. D. VIII. All substance is necessarily infinite. Dem. There exists but one substance of the same attribute; and it must either exist as infinite or finite. But not finite, for (per def. two) as finite it must be limited by another substance of the same nature, and in that case there would be two substances of the same attributes, which (per prop, five) is absurd. Substance therefore is infinite.--Q. E. D.