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Lubavitch DOES eat Shalosh Seudos
THE LUBAVITCH CUSTOM IS TO EAT SHALOSH SEDUOS BUT NOT TO WASH FOR BREAD
In Hayom Yom (22 Adar I) the Rebbe quotes the Rebbe Rasha”b (Meheirah Yishamah 5648) as saying, “The obligation to eat Shalosh Seudos is learned from the Passuk “Hayom Lo”, therefore we don’t need to eat bread, but one must taste something. As Rabbi Yossi said (Talmud, Shabbos 118b) “May my portion be among those who eat Shalosh Seudos”.
So Minhag Chabad is to make sure to taste something for Shalosh Seudos, and the reason why we don’t wash for bread is because of the Posuk “Hayom Lo”.
What this means is as follows: The Gemarah in Shabbos learns out the obligation to eat 3 meals of Shabbos from the following Posuk: “And Moshe said, eat [the Monn] today (Hayom), because today (Hayom) is a Shabbos for Hashem, today you will not (Hayom Lo) find [the Monn] in the field”. Since the Pasuk says the word “Hayom” 3 times (when speaking about the Monn which served them as bread), we learn that one must eat 3 meals on Shabbos.
Based on the words of Rabbeinu Bechayei (one of the greatest students of the Rashb”a who lived over 700 years ago) The Tzemach Tzedek rules (Piskei Dinim Chiddushim on Rabbeinu Yerucham p. 357) that since this meal is learned from the words “Hayom Lo”, unlike the first two meals which are learned from the word “Hayom”, we don’t need to eat bread for Shalosh Seudos, unlike the first two meals.
Similar interpretations are found in the Baal Halevushim (Orach Chaim 291:5) and in the Chachmas Shelomo (ibid).
The concept of not washing of Shalosh Seudos is not a radical concept, and certainly was not invented by Chabad:
The Tur brings down the opinion of some of the Rishonim who hold that one can eat “Minei Targimah” for Shalosh Seudos, and not necessarily bread. Some commentaries (Ra”n, Rash”i) render “Targimah” as fruits, while others (Ros”h) render it as pastries. The Bac”h explains that since one eats a full meal in the morning, one can satiate himself with “Minei Targimah” and it is considered as if he has actually eaten bread. Based on this, the Shulchan Aruch writes: “Some say you can fulfill Shalosh Seudos with any food which is made from the five types of grain, and some say that even fruits are sufficient.”
So the statement of Sally that “Lubavitch also does not eat Seudah Shelishit, which is an Halachic obligation, and was very fundamental to Chasidim all over. This also is a clear violation of Halachah in Lubavitch, and a break away from traditional Chassidus” is pure nonsense. The custom of Lubavitch has a source in the works of the Rishonim and Poskim, and in the writing of Rabbeinu Bechayei, the Levush, the “Chochmas Shelomo” and the “Tzemach Tzedek”.
(And even though the Shulchan Aruch writes that it is good to be machmir and eat bread like the opinions that one must eat bread even for Shalosh Seudos, it is not because he determined that those opinions are move valid than the ones that hold that you should wash, rather, since there is doubt which opinion we should follow, is better to wash so that we be on the safe side. For this reason the Halachah is that if one forgets to recite Retzei by Shalosh Seudos he does not repeat the Bentching (unlike the first two meals), because the entire meal is not obligatory according to many opinions, so to be on the safe side and not risk saying a Brachah Levatalah we don’t repeat the bentching (See Shulchan Aruch and Bac”h Hilchos Birkas Hamazon 188:

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Now, since according to Kabbalah it is better not to eat bread (see following posts), we follow the opinions that fruits or cakes are sufficient.