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Definition D5520
Closed real interval tagged partition

Let $[a, b] \subset \mathbb{R}$ be a D544: Closed real interval such that
 (i) $$a < b$$
An D548: Ordered pair $P = ((x_0, x_1, \ldots, x_N), (x^*_1, \ldots, x^*_N))$ is a tagged partition of $[a, b]$ if and only if
 (1) $$a = x_0 < x_1 < \cdots < x_N = b$$ (2) $$\forall \, n \in \{ 1, \ldots, N \} : x_{n - 1} \leq x^*_n \leq x_n$$

Let $[a, b] \subset \mathbb{R}$ be a D544: Closed real interval such that
 (i) $$a < b$$
An D548: Ordered pair $P = ((x_0, x_1, \ldots, x_N), (x^*_1, \ldots, x^*_N))$ is a tagged partition of $[a, b]$ if and only if
 (1) $$a = x_0 < x_1 < \cdots < x_N = b$$ (2) $$\forall \, n \in \{ 1, \ldots, N \} : x^*_n \in [x_{n - 1}, x_n]$$