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I think that if $H$ has $D$ as a subroutine, when $D$ is running, $D$ could terminate in a halting state, making the whole of $H$ terminate without exiting the subroutine. I think this is the core of your argument - stated again here: $D$ can be built so that, if needed, it directly terminates in such necessary state without passing through the rest of $H$’...


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