This satisfies all the required conditions except Since for large enough, this can be achieved by replacing by Since off it follows from a classical theorem of Whitney [13, Theorem III] (with in that theorem) that there is a function on of class in such that, if then if and if.
Since the law of contradiction is a classical theorem, intuitionistic logic is contained in classical logic.
By the classical theorem of Ehresmann [148], deformation equivalent varieties are diffeomorphic, and moreover, via a diffeomorphism carrying the canonical class to the canonical class.
Then, by applying a classical uniqueness theorem to the domain Ω j, ∑ h = 1 n ( n + 1 ) ∕ 2 c h j v h ( x ) = 0, x ∈ Ω j, j = 1, …, m, from which it easily follows that c h j = 0, h = 1, …, n ( n + 1 ) / 2, j = 1, …, m.
To construct such a sequence, just take a unitary continuous linear functional in a real, nonreflexive Banach space which fails to be norm-attaining on the closed unit ball of (the existence of such an is guaranteed by a classical, well-known theorem of James, cf. [8]), and define (4.7).
Clearly, Theorem 1.11 is a generalization of the classical theorem of multivalued contractive mappings (Nadler [21]).
Proof planning is a technique for theorem proving which replaces the ultra-efficient but blind search of classical theorem proving systems by an informed knowledge-based planning process that employs mathematical knowledge at a human-oriented level of abstraction.
Remark 3.2 We point out that the same statement of above given result can be obtained by using a classical direct methods theorem (see [14]), but in addition we get the location of the solution, hence in particular the solution is nontrivial.
A classical convergence theorem of nonexpansive mappings has been established by Xu and Ori [23].
convergent on [ 0, 1 ] ∖ A. By a classical method (using the Lebesgue dominated convergence theorem) it can be shown that this limit is equal μ F -a.e. to h.
