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Here is an example of the proof of Proposition XV. The black circle is level one, the white circle is level two, and the black square is level three.

Prop. XV: Whatsoever is, is in God, and without God nothing can be, or be conceived.

• Proposition XIV: Besides God no substance can be granted or conceived.
  • Def. VI: God is infinite, that is to say, substance consisting of infinite attributes.
  • Prop. XI: God necessarily exists.
    • proof for this: up to the reader.
  • So, if there were any other substances it would have to be explained by an attribute of God.
  • If there were any other substances besides God, then two different substances would have the same attribute.
  • But by Prop. V: There cannot exist in the universe two or more substances having the same nature or attribute.
    • Prop. IV: If there were two or more distinct substances, they would have to be distinguished from one another by difference of attributes or difference of modifications.
    • If only by the difference of their attributes, it will be granted that there cannot be more than one with an identical attribute.
    • So there cannot be two or more distinct substances having the same attribute that are distinguished from one another by different attributes.
    • Prop. I: Substance is prior by nature to its modifications.
    • Since substance is prior by nature to its modifications, two substances cannot be distinguished by a difference in modifications.
    • So they cannot be different because of different attributes or different modifications.
    • So there cannot exist in the universe two or more substances having the same nature or attribute.
  • So there are no other substances except for God.

• Modes can neither be nor be conceived without substance(Def V).
• So all modes exist in God, since God is the only substance.
• There are only substances or modes, since by Axiom I everything which is, is either in itself or in another.

Therefore, without God nothing can be, or be conceived.