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Let F be a complemented subspace of a nuclear Fréchet space E. If E and F both have (absolute) bases $\left(e_n\right)$ resp. $\left(f_n\right)$, then Bessaga conjectured (see [2] and for a more general form, also [8]) that there exists an isomorphism of F into E mapping $f_n$ to $t_n{e}_{\pi \left(k_n\right)}$ where $\left(t_n\right)$ is a scalar sequence, π is a permutation of ℕ and $\left(k_n\right)$ is a subsequence of ℕ. We prove that the conjecture holds if E is unstable, i.e. for some base of decreasing zero-neighborhoods $\left(U_n\right)$ consisting of absolutely convex sets one has ∃s ∀p ∃q ∀r $li{m}_n\left(d_{n+1}(U_q,U_p)\right)/\left(d_n(U_r,U_s)\right)=0$ where...