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Thermonuclear fusion is still going on in the core of a red giant, but it is a different type of thermonuclear fusion. The center of the core has reached high enough temperature and pressure that it can now burn helium, producing carbon.

3 4He --> 12C

The large amount of energy released by this type of fusion pushes the outer layers away, making a giant star. The expansion of volume of the surface layer causes it to cool, appearing red. Thus a red giant.

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Q: Red giant is a luminous giant star of low or intermediate mass that is in a late phase of its evolution with nuclear fusion going on in a shell outside the core but not in the core itself?
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