In the book "Introduction to the theory of linear nonselfadjoint operators in Hilbert space" by Gohberg and Krein a Bari basis of subspaces $\{\mathfrak{N}_k\}_{k=1}^\infty$ of a Hilbert space $\mathfrak{H}$ is defined as a Schauder basis for which there exists an orthogonal basis $\{\mathfrak{E}_k\}_{k=1}^\infty$ of $\mathfrak{H}$ (i.e. $\mathfrak{H} = \bigoplus_{k=1}^\infty \mathfrak{E}_k$) such that $\{\mathfrak{N}_k\}_{k=1}^\infty$ and $\{\mathfrak{E}_k\}_{k=1}^\infty$ are quadratically close, i.e. $$\sum_{k=1}^\infty ||P_{\mathfrak{E}_k} - P_{\mathfrak{N}_k}||^2 < \infty$$ ($P_{\mathfrak{U}}$ is the orthogonal projection onto $\mathfrak{U}$). On page 343 in remark 5.2 it is claimed that if $\{\mathfrak{N}_k\}_{k=1}^\infty$ is a Bari basis with $\sup_{k \in \mathbb{N}} \dim \mathfrak{N}_k < \infty$, then $$\sum_{\substack{k,l=1 \\ k \neq l}}^\infty \cos(\mathfrak{N}_k,\mathfrak{N}_l)^2 < \infty$$ where $\cos(\mathfrak{U},\mathfrak{V}) = ||P_{\mathfrak{U}}P_{\mathfrak{V}}|| = ||P_{\mathfrak{U}}-P_{\mathfrak{V}^\perp}||$ is the cosine of the angle between $\mathfrak{U}$ and $\mathfrak{V}$.
How can one prove this claim?