Rif Brachot 40a

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40a

{we have digressed to Pesachim 54a, but will shortly return to Brachot 53b}
We bless upon the light, both at the termination of Shabbat and at the termination of Yom Kippur, and so do the people conduct themselves.
They asked {from a Tannaitic source}: We do not bless upon the light except at the termination of Shabbat, since the beginning of its creation was then, and when he sees it he blesses immediately. Rabbi Yehuda says: he orders them upon a cup {of wine}. - And Rabbi Yochanan said: The halacha is like Rabbi Yehuda.
It is not a question. Here is on the flame that rested, and here is on fire that came from wood and stones - that we do not bless at the termination of Yom Kippur except upon fire that rested, but fire from wood and stones, no.

{Brachot 51b}
Mishna:
IF ONE HAS EATEN AND FORGOTTEN TO SAY GRACE,
BETH SHAMMAI SAY THAT HE MUST RETURN TO THE PLACE WHERE HE ATE AND SAY THE GRACE,
WHILE BETH HILLEL SAY THAT HE SHOULD SAY IT IN THE PLACE WHERE HE REMEMBERED.

UNTIL WHEN CAN HE SAY THE GRACE?
UNTIL SUFFICIENT TIME HAS PASSED FOR THE FOOD IN HIS STOMACH TO BE DIGESTED.

IF WINE IS SERVED TO THEM AFTER THE FOOD, AND THAT IS THE ONLY CUP THERE,
BETH SHAMMAI SAY THAT A BLESSING IS FIRST SAID OVER THE WINE AND THEN [THE GRACE] OVER THE FOOD,
WHILE BETH HILLEL SAY THAT A BLESSING IS FIRST SAID OVER THE FOOD AND THEN OVER THE WINE.

ONE SAYS AMEN AFTER A BLESSING SAID BY AN ISRAELITE BUT NOT AFTER A BLESSING SAID BY A CUTHEAN, UNLESS THE WHOLE OF IT HAS BEEN HEARD.

Gemara:
{Brachot 53b}:
Rav Zevid said: They only argue in the case where one forgot, but if he omitted wilfully he must return to his place and say grace.

There were once two disciples who omitted to say grace. One who did it accidentally followed the rule of Bet Shammai and found a purse of gold, while the other who did it purposely followed the rule of Bet Hillel, and he was eaten by a lion.

{The Mishna had said:} "UNTIL WHEN CAN HE SAY THE GRACE? UNTIL SUFFICIENT TIME HAS PASSED FOR THE FOOD IN HIS STOMACH TO BE DIGESTED.":
And how much is the time of digestion?
Rabbi Yochanan said: As long as he is still hungry from that eating.

"ONE SAYS AMEN AFTER A BLESSING SAID BY AN ISRAELITE":
They learnt {in a brayta - in our gemara, this is a statement of Rav}: We respond Amen after everyone except children in school, because they are merely learning.
This is the case only when it is not the time for them to say the haftarah; but when it is the time for them to say the haftarah, we respond Amen after them.

Rav Yehuda cited Rav, and some say it was taught in a brayta: Vayikra 11:44:

מד כִּי אֲנִי ה, אֱלֹקֵיכֶם, וְהִתְקַדִּשְׁתֶּם וִהְיִיתֶם קְדֹשִׁים, כִּי קָדוֹשׁ אָנִי; וְלֹא תְטַמְּאוּ אֶת-נַפְשֹׁתֵיכֶם, בְּכָל-הַשֶּׁרֶץ הָרֹמֵשׂ עַל-הָאָרֶץ.

44 For I am the LORD your God; sanctify yourselves therefore, and be ye holy; for I am holy; neither shall ye defile yourselves with any manner of swarming thing that moveth upon the earth.

וְהִתְקַדִּשְׁתֶּם - 'sanctify yourselves' - this refers to the first waters {washing before the meal}.
וִהְיִיתֶם קְדֹשִׁים - 'and be ye holy' - this refers to the last waters {washing after the meal}
כִּי קָדוֹשׁ - 'for ... holy' - this refers to the good oil {with spices which oil spreads on his hands after the meal}