E3P45D
Demonstration — Part III
Latin
Nam res amata eum qui ipsam odit, odio contra habet (per propositionem 40 hujus) adeoque amans qui aliquem imaginatur rem amatam odio habere, eo ipso rem amatam odio hoc est (per scholium propositionis 13 hujus) tristitia affectam esse imaginatur et consequenter (per propositionem 21 hujus) contristatur idque concomitante idea ejus qui rem amatam odit tanquam causa hoc est (per scholium propositionis 13 hujus) ipsum odio habebit. Q.E.D.
English (Elwes 1883)
The beloved object feels reciprocal hatred towards him who hates it (III. xl.); therefore the lover, in conceiving that anyone hates the beloved object, conceives the beloved thing as affected by hatred, in other words (III. xiii.), by pain; consequently he is himself affected by pain accompanied by the idea of the hater of the beloved thing as cause; that is, he will hate him who hates anything which he himself loves (III. xiii. note). Q.E.D.
Modern English
The beloved thing hates in return the one who hates it (E3P40). So the lover who imagines someone to be affected with hatred toward the beloved thing imagines the beloved thing to be affected with hatred, that is (E3P13S) with sadness, toward that person. Consequently the lover is himself affected with sadness (E3P21), accompanied by the idea of the one who hates the beloved thing as its cause. That is (E3P13S), he will hate that person. Q.E.D.