ANSWER: As you are aware, the public prayer requires at the minimum a minyan (lit. "quorum") of ten adult Jewish males. This is derived from one of the hermeneutic principles (middot) of interpretation through which the Torah is expounded by our Sages, namely, the exegetical rules by which halachot are derived from the Biblical text. One of them is gezera shava, the principle according to which a law is inferred from verbal analogy. Thus, if the same word or phrase appears in two separate verses in the Torah, and a certain halacha is explicitly stated in one of them, we may under certain circumstances infer on the basis of "verbal analogy" that the same halacha applies in the second case as well.

The Mishna (Megilla 23b) states, "One may not recite (repeat) the introduction to the Shema [Rashi: For the benefit of those who came late to the synagogue, one may repeat the Kaddish, Barechu, and the first blessing of Keriat Shema], nor does one [the sheliach tzibbur] pass before the Ark, nor do [the priests] lift their hands [to bless the congregation], nor is the Torah read [publicly], nor the haftara read from the Prophets [scroll], nor are pauses made [at funerals], nor is the blessing [of consolation] for mourners said, nor the [formal words of] comfort of mourners, nor the blessing of bridegrooms, nor is the name [of G-d] mentioned in the invitation to say Grace - save in the presence of ten. . ."

The Gemara cites R. Hiyya b. Abba in the name of R. Yochanan, who explains the source of this requirement to be due to the exegesis of a gezera shava involving two verses in the Torah. The first is found in Parashat Emor (Leviticus 22:32), where we are instructed not to violate intentionally any of Hashem's commands, because it would result in a diminution of Hashem's honor: "Velo techallelu et shem kodshi venikdashti betoch Bnei Yisrael, ani Hashem mekaddishchem - You shall not desecrate My holy Name, and I will thus be sanctified in the midst of (betoch) the Children of Israel; I am Hashem who sanctifies you." We must zealously strive to sanctify Hashem, and in so doing we ourselves will be sanctified.< BR>
The second verse is in Parashat Korach (Numbers 16:21), when Hashem instructs Moshe and Aharon to separate themselves from the rebellious Korach and his followers: "Hibbadlu mitoch ha'edah hazot va'achaleh otam ke'raga - Separate yourselves from amid (mitoch) this congregation, and I shall destroy them in an instant."

This leads us to another gezera shava that is drawn between the word "ha'edah" in this verse and a verse in Parashat Shelach regarding the episode of the spies whom Moshe had sent to investigate the land of Canaan. When they returned with a bad report about the land that had been promised to the Children of Israel, Hashem made His wrath known to Moshe and Aharon (Numbers 14:27): "Ad matai la'edah hara'ah hazot asher hema mallinim alai, et telunot Bnei Yisrael asher hema mallinim alai shamati - How long [shall I bear] this evil congregation that complains against Me? I have heard the complaints of the Children of Israel against Me." Thus, just as there were ten spies, so do we require a quorum of ten people (to form a "congregation").

Though we find no reference to the situation of our discussion in the Babylonian Talmud, we do find the following in the Jerusalem Talmud (Megillah 4:4) relating to our Mishna: "One may not recite the introduction to the Shema with fewer than ten [men]. However, if they started with ten and some [of them] left, they may conclude. Nor does one [the sheliach tzibbur] pass before the Ark with fewer than ten; however, if they started with ten and some left, they may conclude. Nor do [the priests] lift their hands with fewer than ten; however, if they started with ten and some left, they may conclude. We do not read from the Torah with fewer than ten; however, if they started with ten and some left, they may continue. Nor do we read from the Prophets [scroll] with fewer than ten; however, if they started with ten and a few left, they may conclude."

The Sages indicate their displeasure with such action as can be seen from the following verse (Isaiah 1:28), "... [V]e'ozvei Hashem yichlu - ... those who desert Hashem will be destroyed." This is applied to those who hasten to leave in the middle of the prayer services, thus diminishing the number of those who form the quorum.

Based on this Gemara, the Mechaber (Orach Chayyim 55:23) rules that in all cases that require a minimum of ten, if they began to say Kaddish or Kedusha with the participation of ten adult males, and a few left, they can finish the prayer, provided a majority has remained.

Rema adds, based on the Gemara in the Jerusalem Talmud, that it is sinful to leave, citing the above referenced verse in Isaiah. However, it is permitted to leave as long as ten men will remain, in which case the verse in Isaiah does not apply to such individuals.

The Mechaber clarifies (55:3) that "if they began Avot [the sheliach tzibbur began to recite the Amida] and a few left, they may finish, and even recite the Kedusha [subsequent to their leaving]."

Rema adds, "If a few left after they started to pray 'Yotzer or' [the beginning of Birkat Keriat Shema], the sheliach tzibbur shall not start to say the Amida [Chazarat HaShatz] in a loud voice since Birkat Keriat Shema was already concluded." Thus the Amida is treated as a separate tefilla.

It is obvious that Rema indicates that there was no quorum at that point, and they might want the sheliach tzibbur to recite a "hoiche (loud) Kedusha" (infra 124:2, Rema). Rema continues: "If they left after he began to recite Chazarat HaShatz and Kedusha [the Mishna Berura ad loc. notes that this applies even if Kedusha had not been recited yet], they may complete the entire Kedusha [and the rest of the Amida] and recite the whole Kaddish afterwards, as that [Kaddish] is connected to the Amida, since he recites "Titkabbel Tzelot'hon - May the prayers ... be accepted." However, continues Rema, they may not read from the Torah [on Shabbat, Monday or Thursday], as that is considered a separate, unconnected matter.

As regards Ma'ariv, Rema rules that the subsequent Kaddish is not connected to Birkat Keriat Shema. Therefore, if they started Birkat Keriat Shema with ten men and some left, the Taz - cited by the Mishna Berura - rules that he may only recite the Kaddish at the conclusion of Birkat Keriat Shema, but not the Kaddish after the Amida, as that Amida is not considered tefilla betzibbur (a congregational prayer).

When we say that a "few" left, it means that a majority of the ten remain. As to what constitutes a majority, both the Aruch HaShulchan and the Mishna Berura cite the Pri Megadim, who rules that if six of the original ten remain, that is considered sufficient for completing the prayer.

In our case, where the individuals left during Chazarat HaShatz of Shacharit, the sheliach tzibbur can say the half Kaddish after the Shemoneh Esreh, followed by Ashrei, U'va LeTziyyon, and Kaddish Titkabbel. However, Aleinu and Shir shel Yom and their Kaddish recitals will not be said by the sheliach tzibbur. Likewise, at Mincha and Ma'ariv he would not say Aleinu and its concluding Kaddish.

We see that each segment of the prayer service is treated as a separate unit and the mere fact that there were ten men at the beginning does not allow us to continue the tefilla betzibbur beyond that particular segment.

The Mechaber (ad loc.) cites the views of those who allow, in an extreme situation, to include, with nine adults, a minor above age six who understands to whom he is praying.

Rema disagrees and says that even when the child has a Chumash in his hand, he should not be included to complete a quorum. However, there are those who are lenient in an extreme case (Rosh, Mordecai, Hagahot Maimoniyot).

Shulchan Aruch HaRav (ad loc.) likewise notes that we do not protest if some people do so, as they have upon whom to rely (the above mentioned authorities), whether the minor child is holding a Chumash in his hand or not.

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