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The recurrence theorem of the Frenet formulas  Uploading ....Studying the Frenet formulas I have concluded that they are recursive. More specifically, using the trigonometric form of the Frenet formulas, we proved the following
Theorem: If there is a right trihedron of the n order 
that satisfies the Frenet formulas of the n order, written in the trigonometric form
then there is still a right trihedron of the n order
that satisfying, in turn, the Frenet formulas of the n+1 order written also in the trigonometric form
where
 and 
.
Demonstration: Through relations  and 
we have that
so
.
We also have
whence
.
Now, we derive the unit vectors of the trihedron of the n+1 order
and we obtain
.
Replacing
and
, we obtain
.
But, from the definition of the unit vectors of the high order, we know that
,
so
.
Because  and  ,
finally result that
,
qed.
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