Valmiki Ramayana Ayodhya Kanda Sarga 9 Shloka 33
Original Shlokaएवं प्रव्राजितश्चैव रामोऽरामो भविष्यति। भरतश्च हतामित्रस्तव राजा भविष्यति।।2.9.33।।
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एवम् thus, प्रव्राजित: exiled, राम: Rama, अराम: will no longer cause any delight to people, भविष्यति will become, भरतश्च Bharata also, राजा as king, हतामित्रश्च with enemies destroyed, भविष्यति will become.
Thus, if Rama is exiled he will no longer be Rama (be able to cause delight to the people). Bharata will become king with his enemy subdued.