  H. Cartan in his book on differential calculus proved a theorem generalizing
a Cauchy's mean-value theorem to the case of functions taking values in a
Banach space.
  Cartan used this theorem in a masterful way to develop the entire theory of
differential calculus and theory of differential equations in finite and
infinite dimensional Banach spaces.
  The author proves a generalization of this theorem to the case when the
inequality involving the derivatives holds everywhere with exception of a set
of Lebesgue measure zero, and the derivatives are replaced by weaker
derivatives. Namely the right-sided Lipschitz derivative and lower right-sided
Dini derivative, respectively.
  He also presents applications of the theorem to the study of Lipschitzian
operators in Banach spaces. Lipschitzian operators played pivotal role in the
n-body problems of electrodynamics, as also in general n-body problem of
Einstein's special theory of relativity. For references see Bogdan
arXiv:0909.5240 and arXiv:0910.0538.
  Using the generalization of Cartan's theorem the author proves a version of
the fundamental theorem of calculus in a class of Bochner summable functions.
In the process he introduces the reader to the generalized theory of
Lebesgue-Bochner-Stieltjes integral and Lebesgue and Bochner spaces of summable
functions as developed by Bogdanowicz. \cite{bogdan10}--\cite{bogdan23}.
