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The quotient space is the set of the equivalence class for this relation in p is the tangeant space in p. The quantity [tex](x o \gamma)'(0)[/tex] for a certain curve permit a bijection from the tangent space to [tex]\mathbb{R}^{n}[/tex]. So the tangent space is a vectorial space. And a vector of this space is a an equivalences class. A set of curve which have the same [tex](x o \gamma)'(0)[/tex]

How to demosntrate that this definition is independant of the choice of card please?

I try to say that [tex](x o \gamma_{1})'(0) = (x o \gamma_{2})'(0) \Leftrightarrow (x o y^{-1} o y o \gamma_{1})'(0) = (x o y^{-1} o y o \gamma_{2})'(0)[/tex] but I don't know how to go on.

Thank you in advance and have a nice afternoon:D.