Shulchan Arukh, Orach Chayim Siman 140
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הקורא בתורה ונשתתק העומד יתחיל ממקום שהתחיל הראשון ויברך בתחל' ובסוף ולהרמב"ם לא יברך בתחלה [הגה] ואפי' בזמן הזה שש"ץ קורא דינא הכי [הר"י פרק אין עומדין]:
One who was reading from the Torah and became silent [i.e. became unable to continue], the one who stands as a replacement should start from the place where the first one began, and should bless before and after [reading]. And according to the Rambam, should not make the blessing before. Gloss: And even nowadays where the prayer leader reads, the law is the same. (The R"i - chapter "Ein Omdin" [B'rakhot, Chapter 5)
העומד לקרות בתורה וברך ברכה שלפניה וקרא מקצת פסוקים ופסק ודבר דברי תורה או דברי חול לא הוי הפסק ואינו צריך לחזור ולברך:
One who stood to read from the Torah, and blessed before it, and read some of the verses, and then stopped and spoke [either] words of Torah or mundane words, this is not [considered] an interruption, and [the reader] does not need to go back and bless [the blessing before once again and start over].
העולה לקרות בתורה והראו לו מקום שצריך לקרות וברך על התורה והתחיל לקרות או לא התחיל והזכירוהו שפרשה אחרת צריך לקרות וגלל הספר תורה למקום הצריך לקרות בו יש אומרים שאינו צריך לחזור ולברך וי"א שצריך:
One who went up to read from the Torah, and they showed [the person] the place where [that person] should read from, and [the person] blessed over the Torah and started to read or [even] did not start to read, and then they reminded [the person] that [the person] is supposed to read a different parasha [section of the reading], and [then the person] rolled the Torah scroll to the place [the person] is supposed to read: there are those who say that [the person] does not need to go back and bless [again], and there are those who say there is a need [to do so].