Retain the two given logical implications and temporarily assume their common starting proposition.
1  P → Q                    premise
2  Q → R                    premise
3  | P                      assume
4  | Q                      →-elim, 3 and 1
5  | R                      →-elim, 4 and 2
6  P → R                    →-intro, discharge 3–5
Only the original premises remain undischarged, giving the required composition of logical implications.

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