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I need help with an aspect of the proof of Theorem 3.3.14 ... ...

Theorem 3.3.14 reads as follows:

Can someone please explain/demonstrate explicitly why \(\displaystyle (e_1)_p, (e_2)_p, \ ... \ ... \ , (e_{n-1} )_p\) are the standard basis vectors for \(\displaystyle T_p( S_f )\) ...

Help will be much appreciated ... ...

Peter

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The above post mentions Theorem 3.3.13 so I am providing text of the theorem together with a relevant definition ... as follows:

Hope that helps readers follow the post ...

Peter