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(** *                     Definition of  Rpower                   *)
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Definition Rpower (x y:R) := exp (y * ln x).
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(** *                     Properties of  Rpower                   *)
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(** Note: [Rpower] is prolongated to [1] on negative real numbers and it thus does not extend integer power. The next two lemmas, which hold for integer power, accidentally hold on negative real numbers as a side effect of the default value taken on negative real numbers. Contrastingly, the lemmas that do not hold for the integer power of a negative number are stated for [Rpower] on the positive numbers only (even if they accidentally hold due to the default value of [Rpower] on the negative side, as it is the case for [Rpower_O]). *)

Theorem D_in_imp :
forall (f g:R -> R) (D D1:R -> Prop) (x:R),
(forall x:R, D1 x -> D x) -> D_in f g D x -> D_in f g D1 x.