"Tefila Besibur" congregational prayer requires the presence of ten men. The presence of ten men is required to allow reciting certain parts of the prayer service, such as Kaddish and Barechu, and to receive the special benefits of praying with a Minyan.
If ten men are present, but several of them have already prayed and are therefore not currently praying, the group may still recite Kaddish, Barechu and other parts of the prayer service requiring a Minyan. As long as at least six of the men are currently praying, these parts of the service may be recited. However, Rav Moshe Feinstein (Russia-New York, 1895-1986), in his Iggerot Moshe, writes that the Amida prayer recited by less than ten men does not qualify as a public prayer service. He notes that the Gemara in Masechet Shabbat (6) speaks of the Shechina (Divine Presence) residing among ten people who are praying indicating that this applies only if all ten are praying at the same time. Therefore, in his view, even though Kaddish and Barechu may be recited if four of the ten men have already prayed, the Amida prayer recited under such circumstances does not qualify as a public prayer service, and the people do not access the special benefits of public prayer.
Hacham Ovadia Yosef, however, in his work Yehave Daat (vol. 5, Siman 7), disagrees with this ruling. He proves from the comments of the Rambam (Rabbi Moshe Maimonides, Spain-Egypt, 1135-1204) in Hilchot Tefila that a prayer recited in such a situation, where only six of the ten men present are currently praying, indeed qualifies as a public prayer. As such, they receive all the benefits of public prayer, the Hazan can recite the repetition of the Amida, and the Kohanim recite Birkat Kohanim. In short, as long as ten men are present and at least six of them are currently praying, they can pray a full public prayer service, just as though there were ten praying. This is also the ruling of Rav Yishak Weiss (1902-1989), in his Minhat Yishak.
Summary: The presence of ten men qualifies as a Minyan as long as at least six of them are currently praying, even though the other four have already prayed. If this condition is met, they may conduct an ordinary, complete public prayer service, including the repetition of the Amida with Birkat Kohanim, Kaddish and Barechu.
