  The notion of deduction in  LA is defined in the customary way; if Δ is a set of sentences of  LA and φ ∈  LA, then a deduction of φ from Δ in  LA is a deduction of φ from Δ in  L(ω1, ω) every formula of which is in  LA. We say that φ is deducible from Δ in  LA if there is a deduction D of φ from Δ in  LA; under these conditions we write Δ  ⊢A φ. In general, D will not be a member of A; in order to ensure that such a deduction can be found in A it will be necessary to impose further conditions on A. 