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By virtue of the cycle, any of these bracketed representations is actually infinite. For example, ; which is to say,

Conversely, numbers to the left of the bracket can be eliminated, if the gap to the next element is just .

More formally,

If , then also .

The converse is also true.

If is given by

multiplying by up and down, we get

then adding will produce

where the right-hand side expression corresponds to the alternative bracketed expression for we were looking for, .

The proof can be traced backwards, in order to show that the implication also holds in the other direction, making it possible to remove from the front of the brackets.