Recursive Set Challenge

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we knoe enter image description here

also we know enter image description here

for example if A be any arbitrary r.e set. can we always Necessarily the following is TRUE ? (always) any description is good. (bar sign means complement)

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Every r.e. set is many-one reducible to the halting set $K$. That fact, together with the theorem you cited, immediately implies an affirmative answer to your question.