In this paper we study the nature and properties of those sequences \({\mathcal S}=(a_ n)_{n\in N}\) in a Banach space B which verify one of the following conditions: 1) \([a_ k;k\in S]\cap [a_ r;r\in T]=[a_ h;h\in S\cap T]\) for any finite \(S\subset N\) and infinite, with infinite complement, subset \(T\subset N.\) 2) \([a_ k\); \(k\in S]\cap [a_ r;r\in T]=[a_ h;h\in S\cap T]\) for any cofinite \(S\subset N\) and any infinite, with infinite complement, subset \(T\subset N.\) Taking into account former results of \textit{A. Plans} and the author [see ibid. 40, 452-458 (1983; Zbl 0517.46003)] we have a complete classification of the sequences in a Banach space in terms of intersections of closed linear spans of subsystems of the given sequence.