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

On Henri Cartan's vectorial mean-value theorem and its applications to  Lipschitzian operators and generalized Lebesgue-Bochner-Stieltjes integration  theory