Are objects with the same id always equal when comparing them with ==?

If I have two objects o1 and o2, and we know that

id(o1) == id(o2)

returns true.

Then, does it follow that

o1 == o2

Or is this not always the case? The paper I'm working on says this is not the case, but in my opinion it should be true!

Not always:

>>> nan = float('nan')
>>> nan is nan
True

or formulated the same way as in the question:

>>> id(nan) == id(nan)
True

but

>>> nan == nan
False

NaN is a strange thing. Per definition it is not equal nor less or greater than itself. But it is the same object. More details why all comparisons have to return False in this SO question.

The paper is right. Consider the following.

class WeirdEquals:
    def __eq__(self, other):
        return False

w = WeirdEquals()
print("id(w) == id(w)", id(w) == id(w))
print("w == w", w == w)

Output is this:

id(w) == id(w) True
w == w False