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Let R be a commutative Noetherian ring. Let and be ideals of R and let N be a finitely generated R-module. We introduce a generalization of the -finiteness dimension of $f_{}^{}\left(N\right)$ relative to in the context of generalized local cohomology modules as $f_{}^{}(M,N):=infi\ge 0\left|\subseteq \surd \right(0{:}_R{H}_{}^i(M,N))$, where M is an R-module. We also show that $f_{}^{}\left(N\right)\le f_{}^{}(M,N)$ for any R-module M. This yields a new version of the Local-Global Principle for annihilation of local cohomology modules. Moreover, we obtain a generalization of the Faltings Lemma.