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Ya. the exact meaning of the word “same category of school” is lacking in the rule but I think the Secondary and Higher secondary schools are of different categories. Before taking any kind of steps you referred, I would like to suggest you to find a person who is nearest and dearest to that head teacher and seek help. Disclose him/her the whole and make it assure that your wife is willing to teach the H.S. Students and she will also undertake the matter in writing if required. Ignore this if you already taken this kind of steps. Be waiting to see the future with best wishes to you dear friend.