Prove that $|2^{N^{+}}||2^{N^{+}}{2}^{N^{+}}|$and $|2^{N^{+}}{2}^{N^{+}}||2^{N^{+}}|,$ to conclude that $|2^{N^{+}}|=|2^{N^{+}}{2}^{N+}|.$ Here you can think of $2^{N^{+}}$as the set of all infinite binary sequences, two of which are equal if and only if every digit in the sequence matches.