Let \(X\) be a separable Banach space. There are given various definitions of bases \((x_n)\) in \(X\), where \(x_n\in X\) for \(n= 1,2,\dots\)\ . Among else there is defined the uniformly minimal basis \((x_n)\) with quasi-fixed brackets and permutations, denoted in the following by \((D_4)\). The paper is devoted to proofs of the following fundamental two theorems: 1) Every separable Banach space has a \((D_4)\)-basis. 2) Every subspace of a separable Banach space \(X\) has a \((D_4)\)-basis which can be extended to a \((D_4)\)-basis of \(X\).