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On the Decomposition of Finite Sequences

For simplicity we adopt the following rules: f1, f2, f3 will denote finite sequences, p, p1, p2, p3 will be arbitrary, f will denote a finite sequence, and i, k will denote natural numbers. Next we state a number of propositions: (3) For every set X and for every i such that X ⊆ Seg i and 1 ∈ X holds (SgmX)(1) = 1. (4) For every finite sequence f such that k ∈ dom f and for every i such that 1 ≤ i and i < k holds f(i) 6= f(k) holds f(k) f = k.