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 ושבעה עשר בתמוז הפך לנו לשמחה ולששון*

The  Brisker Rav said in the name of his father R. Chaim,  “when Chazal established the four fasts, It was not that they established the fasts and then exempted the חולה (sick) from fasting.

Some Poskim** add that according to R.Chaim if חולים are part of and are needed to complete the Minyan then you might not be allowed to Lain ויחל since there is no Minyan of Yidden who are participating in the fast even if all ten are fasting.  

Mori V’rabi Harav Hagaon R. Shlomo Miller Shlita said in jest that perhaps we can add one more Brisker Chumra, “not to give an Aliya on a fast day to anyone who might be a חולה.”