By Ladoshkin M.V.

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**Example text**

T =t Ri (t1 , . . ,t , . . ,t λ (i) ) → Ri (t1 , . . ,t , . . ,t λ (i) ) (Ri ) . t =t . f j (t1 , . . ,t , . . ,tμ ( j) ) = f j (t1 , . . ,t , . . 18) We leave the proofs of (S), (Tr) and (fj ) to the reader; we now present that of (Ri ). We shall carry out the replacement of t by t in the first argument of Ri . It will be clear that we shall be able to carry out the replacement of every arbitrary argument . of Ri by the same method. Thus we assume that we are given a proof of t = t : ..

Let us say ϕ is ϕn . From the consistency of Σ ∗ ∪ {ϕ } follows that of Σ ∗ ∪ {ϕn } and, a fortiori, that of Σn ∪ {ϕn }. 1). From this follows ϕn ∈ Σ ∗ . Thus Σ ∗ is maximal (as well as consistent). 2, in order to obtain a maximal consistent extension Σ ∗ ⊆ Sent(L ) of Σ . For such a Σ ∗ we have: (I) (II) Σ ∗ is maximal consistent in Sent(L ); for each ∃x ϕ in Sent(L ) there is a k ∈ K with (∃x ϕ → ϕ (x/ck )) in Σ ∗ . These two properties of Σ ∗ canonically determine a domain in which all sentences σ ∈ Σ ∗ hold, as we shall see.

2, in order to obtain a maximal consistent extension Σ ∗ ⊆ Sent(L ) of Σ . For such a Σ ∗ we have: (I) (II) Σ ∗ is maximal consistent in Sent(L ); for each ∃x ϕ in Sent(L ) there is a k ∈ K with (∃x ϕ → ϕ (x/ck )) in Σ ∗ . These two properties of Σ ∗ canonically determine a domain in which all sentences σ ∈ Σ ∗ hold, as we shall see. In particular, all σ ∈ Σ will hold there. We first consider the set of constant L -terms: CT := {t ∈ Tm(L ) | no variable occurs in t }. 2) CT contains, in particular, all ck with k ∈ K .

### A-modules over A-algebras and Hochschild cohomology for modules over algebras by Ladoshkin M.V.

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