>>>>>Proof--The mind, as far as possible, endeavours to conceive those things which increase or help the body's power of activity (III. xii.); in other words (III. xii. note), those things which it loves. But conception is helped by those things which postulate the existence of a thing, and contrariwise is hindered by those which exclude the existence of a thing (II. xvii.); therefore the images of things, which postulate the existence of an object of love, help the mind's endeavour to conceive the object of love, in other words (III. xi. note), affect the mind pleasurably; contrariwise those things, which exclude the existence of an object of love, hinder the aforesaid mental endeavour; in other words, affect the mind painfully. He, therefore, who conceives that the object of his love is destroyed will feel pain, &c. Q.E.D. XX. He who conceives that the object of his hate is destroyed will also feel pleasure. >>>>>Proof--The mind (III. xiii.) endeavours to conceive those things, which exclude the existence of things whereby the body's power of activity is diminished or constrained; that is (III. xiii. note), it endeavours to conceive such things as exclude the existence of what it hates; therefore the image of a thing, which excludes the existence of what the mind hates, helps the aforesaid mental effort, in other words (III. xi. note), affects the mind pleasurably. Thus he who conceives that the object of his hate is destroyed will feel pleasure. Q.E.D. XXI. He who conceives, that the object of his love is affected pleasurably or painfully, will himself be affected pleasurably or painfully; and the one or the other emotion will be greater or less in the lover according as it is greater or less in the thing loved. >>>>>Proof--The images of things (as we showed in III. xix.) which postulate the existence of the object of love, help the mind's endeavour to conceive the said object. But pleasure postulates the existence of something feeling pleasure, so much the more in proportion as the emotion of pleasure is greater; for it is (III. xi. note) a transition to a greater perfection; therefore the image of pleasure in the object of love helps the mental endeavour of the lover; that is, it affects the lover pleasurably, and so much the more, in proportion as this emotion may have been greater in the object of love. This was our first point. Further, in so far as a thing is affected with pain, it is to that extent destroyed, the extent being in proportion to the amount of pain (III. xi. note); therefore (III. xix.) he who conceives, that the object of his love is affected painfully, will himself be affected painfully, in proportion as the said emotion is greater or less in the object of love. Q.E.D.