I'm reading the book A Course in Metric Geometry by Dmitri Burago, Yuri Burago Sergei Ivanov. In page 45 it says "it is easy to see that all parametrization of a curve have equal length", but I do not understand why this is true.
2.1.1. Definition of length structures.
A length structure on a topological space $ X $ is a class $ A $ of admissible paths, which is a subset of all continuous paths in $ X $, together with a map $ L : A \rightarrow \mathbb{R}^{+} \cup \{\infty \} $; the map is called length of path. The class $ A $ has to satisfy the following assumptions :
$(1)$ The class $ A $ is closed under restrictions : if $ \gamma : [a, b] \rightarrow X $ is an admissible path and $ a \leq c \leq d \leq b$, then the restriction of $ \gamma $ to $ [c, d] $ is also admissible.
(2) $ A $ is closed under concatenations (products) of paths. Namely, if a path $ \gamma : [a, b] \rightarrow X $ is such that its restrictions $ \gamma_1, \gamma_2 $ to $ [a, c] $ and $ [c, b] $ are both admissible paths, then so is $\gamma $.
$(3)$ $ A $ is closed under (at least) linear reparameterizations : for an admissible path $ \gamma : [a, b] \rightarrow X $ and a homeomorphism $ \phi : [c, d] \rightarrow [a, b] $ of the form $ \phi(t) = \alpha t + \beta $, the composition $ \gamma \circ \phi(t) $ is also an admissible path.
We require that $ L $ possesses the following properties :
$(1)$ Length of paths is additive : $ L(\gamma {\big|}_{[a,b]}) = L(\gamma {\big|}_{[a,c]}) + L(\gamma{\big|}_{[c,b]}) $ for any $c \in [a, b] $.
$(2)$ The length of a piece of a path continuously depends on the piece. More formally, for a path $ \gamma : [a, b] \rightarrow X $ of finite length, denote by $ L(\gamma, a, t) $ the length of the restriction of $ \gamma : [a, b] \rightarrow X $ to the segment $ [a, t] $. We require that $ L(γ, a, \cdot) $ be a continuous function.
$(3)$ The length is invariant under reparameterizations : $ L(\gamma \circ \phi) = L(\gamma) $ for a linear homeomorphism $\phi$.
$(4)$ We require length structures to agree with the topology of $X$ in the following sense : for a neighborhood $U_x$ of a point $x$, the length of paths connecting $x$ with points of the complement of $U_x$ is separated from zero : $$ \inf \{ L(\gamma) : \gamma(a) = x, \gamma(b) \in X \setminus U_x \} > 0.$$
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Definition 2.5.1. An (unparameterized) curve is an equivalence class of the minimal equivalence relation satisfying the following : paths $ \gamma_1 : I_1 \rightarrow X $ and $ \gamma_2 : I_2 \rightarrow X $ are equivalent whenever there exists a nondecreasing continuous map $ \phi $ from $ I_1 $ onto $ I_2 $ such that $ \gamma_1 = \gamma_2 \circ \phi $.
Paths (representatives of an equivalence class) are also called parameterizations of the curve and re-parameterizations of one another.
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I need to prove that $ L(\gamma) = L(\gamma \circ \phi) $ where $ \gamma : I \rightarrow X $ is a path and $ \phi : I' \rightarrow I $ is a nondecreasing continuous map, I think I should use broken line to approximate $ \phi $, but I don't know how to do it.
Thanks in advanced