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LOMOV: I’ll prove to you that they’re mine!
CHUBUKOV: You won’t prove it, my darling —
LOMOV: I shall
CHUBUKOV: Dear one, why yell like that? You won’t prove anything just by yelling. I don’t want anything of yours, and don’t intend to give up what I have. Why should I? And you know, my beloved, that if you propose to go on arguing about it, I’d much sooner give up the Meadows to the peasants than to you. There!
LOMOV: I don’t understand! How have you the right to give away somebody else’s property?
CHUBUKOV: You may take it that I know whether I have the right or not. Because, young man, I’m not used to being spoken to in that tone of voice, and so on. I, young man, am twice your age, and ask you to speak to me without agitating yourself, and all that.
When Chubukov refused to accept Lomov's argument, he became enraged and told Chubukov that Oxen Meadows were his, to which Chubukov calmly answered that he couldn't prove it. However, Lomov asserted once more that he would prove ownership of Oxen Meadows.
So, Chubukov tried to inform Lomov gently that there was no need to yell, as yelling would do nothing. He repeated Natalya's comments that he does not wish to claim anything that is not his and does not want to give away what is his. Moreover, he warned Lomov that if he continued to argue about the land, he would rather give it to his people for free than to him.

Lomov responded to Chubukov, saying, Chubukov did not have the right to give away someone's (Lomov's) property to anybody. Upon hearing this, Chubukov tried to communicate to Lomov that the tone in which Lomov chose to speak to him was annoying him. Furthermore, he also stated that he is twice his age and needs to be spoken to with respect rather than in an agitated tone.
Meaning of difficult words:
Agitate To stir, irritate or anger someone
National Council of Educational Research and Training (2007). The Proposal - Anton Chekov (pp. 142-161). Published at the Publication Division by the Secretary, National Council of Educational Research and Training, Sri Aurobindo Marg, New Delhi.