How can I prove this?

[DG]

Case book 3.1.4 Situation C: F4, for whom the DH is batting, pinch hits or pinch runs for the DH. RULING: The DH position is eliminated for the remainder of the game. However, the starting DH could re-enter as a player but not in the role of DH. If the does re-enter, he must re-enter in the same position in the batting order, replacing F4.

That covers part of your question but not whether F4 can re-enter but seems logical since he has only left the game once and he was a starter.