ΧΑΙΡΕ is "Hail!" as in "Hail, Mother Mary". It is the Ancient Greek greeting!
"ΕΥΨΥΧΕΙ" is imperative from the verb "εὐψυχέω - εὐψυχῶ"= "I have a brave heart". But, when it written on a tomb, it means "Farewell".
So, the translation of "ΧΑΙΡΕ ΚΑΙ ΕΥΨΥΧΕΙ" is: "Hail and Farewell".
Also, in Ancient Greek "ἄξιος" means "worthy" and "ἐστί" is the third person from the verb "εἰμί" = "I am".
So, from the back of the tomb (ΑΞΙΟΣ ΕΣΤΙ), it is like saying"He is (a) worthy (man)", even now in death.
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Vasiliki Baskou, Instructor/Director, https://learn-greek-online.com.