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I was curious if anyone could help or provide a reference for the proof to the following lemma

Lemma:

Let $P_{1}$ be the set of polynomials of the first degree and let

$W = w(x) : w \in C([0,1]), w(0) = w(1) = 0\;\text{and}\;w\mid_{(x_{i-1},x_{i})}$

then $W = V_{0}$ and

$w(x) = \sum_{i=1}^{n-1} w(x_{i}) \phi_{i}(x) \forall w(x) \in W = V_{0}$ To note, $V_{0} = \text{span}(\phi_{1},\dots,\phi_{n})$

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