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.
PROP.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.