Almost Existentially Closed Models in Positive Logic

Tis paper explores the concept of almost positively closed models in the framework of positive logic. To accomplish this, we initially defne various forms of the positive amalgamation property, such as h-amalgamation and symmetric and asymmetric amalgamation properties. Subsequently, we introduce certain structures that enjoy these properties. Following this, we introduce the concepts of Δ -almost positively closed and Δ -weekly almost positively closed. Te classes of these structures contain and exhibit properties that closely resemble those of positive existentially closed models. In order to investigate the relationship between positive almost closed and positive strong amalgamation properties, we frst introduce the sets of positive algebraic formulas E T and Alg T and the properties of positive strong amalgamation. We then show that if a model A of a theory T is a E T + ( A ) -weekly almost positively closed, then A is a positive strong amalgamation basis of T , and if A is a positive strong amalgamation basis of T , then A is Al T + ( A ) -weekly almost positively closed.


Introduction
Te notion of strong amalgamation base in the framework of the general model theory was defned by Bacsich in ( [1,2]).He proved that every strong amalgamation basis of a universal theory T with the amalgamation property is algebraically closed in the sense of Jonsson ([3]) and Robinson ([4]).In ( [5]), Eklof has shown that the converse is true in general.He established necessary conditions for members of a special class K to be strong amalgamation basis, even when K is not an elementary class.Tese conditions are expressed in terms of a strong notion of algebraically closed structures, introduced in ([5]), and utilizing the concept of closure operators.
In the conventional models theory, the strong amalgamation property is a characteristic that a structure M can possess within its class of extensions.Tis essentially means that, for any extensions A and B of M, there exists a common extension C of A and B such that A ∩ B � M.
Te positive model theory is concerned essentially with the study of h-inductive theories which are built without the use of the negation.Considering positive formulas and homomorphisms instead of embeddings, positive logic generates new situations beyond the scope of logic with negation.
Consequently, when examining the property of amalgamation, homomorphisms are predominantly utilized instead of embeddings.Terefore, the application of the principle of strong amalgamation mentioned at the end of the frst paragraph of this introduction naturally leads to introduce the concepts of "positive strong amalgamation" and "h-strong amalgamation." In this paper, we will explore one of the specifc aspects of positive logic which embodies the notions of algebraic closedness and strong amalgamation and undertake to study some interactions between these new notions inspired directly or indirectly from the works of Bacsich ( [1]).In the frst section, after summarising the necessary background of the positive model theory, we introduce the general form of symmetric and asymmetric amalgamations.We show that the model completeness of an h-inductive theory can be characterized by a form of symmetric amalgamation.Te second section is devoted to the notions of almost positively closed models and a special class of positive formulas called (A, T)-closed formulas.Note that the terminology "closed formula" here has diferent meaning of the notion of formulas without free variables.We analyse the class of almost positively closed model and present a characterization through some properties of the class of the (A, T)-closed formulas.In the third section, we introduce the notions of positive strong amalgamation and h-strong amalgamation properties.We show that the class of almost positively closed has the positive strong amalgamation property.Furthermore, we give a syntactic characterization of positive strong amalgamation bases.

Basic Defnitions and Notations.
In this subsection, we briefy introduce the basic terminology related to the positive logic.For more details, the reader is referred to ( [6][7][8]).
Let L be a frst order language that contains the symbol of equality and a constant ⊥ denoting the antilogy.Te quantifer-free positive formulas are obtained from atomic formulas using the connectives ∧ and ∨.Te positive formulas are built from quantifer-free positive formulas using the logical operators and quantifer ∧ , ∨ and ∃, respectively.Eventually, the positive formulas are of the form ∃y ϕ(x, y), where ϕ(x, y) is quantifer-free formula.Te variables x are said to be free.Also, a sentence is a formula without free variables.
A sentence is said to be h-inductive if it is a fnite conjunction of sentences of the following form: where φ(x) and ψ(x) are positive formulas.Te h-universal sentences are the negation of positive sentences.
Let A and B be two L-structures.A map h from A to B is a homomorphism if for every tuple a ∈ A and for every atomic formula ϕ; if A ⊨ ϕ(a), then B ⊨ ϕ(h(a)).So, we say that B is a continuation of A.
An embedding of A into B is a homomorphism h: A ⟶ B such that for every a ∈ A and ϕ, an atomic formula, if B ⊨ ϕ(h(a)), then A ⊨ ϕ(a).Te homomorphism h: A ⟶ B is said to be an immersion whenever for every a ∈ A and ϕ a positive formula, if B ⊨ ϕ(h(a)), then A ⊨ ϕ(a).
A class of L-structures is said to be h-inductive if it is closed under the inductive limit of homomorphisms.For more details on the notion of h-inductive sequences and limits, the reader is invited to ( [7]).
Parallel to the role of existentially closed structures in the framework of logic with negation, for every h-inductive theory T, there exists a class of models of T which represent the theory marvellously, and which enjoy the properties desired by every structures; namely, the h-inductive property of the class, the maximality of types (positive formulas satisfed by an element), amalgamation property, and others.Tese modules are called positively closed.
Defnition 1.A model A of an h-inductive theory T is said to be positively closed (in short, pc) if every homomorphism from A to B, a model of T, is an immersion.
Te following lemmas announce the main properties of pc models.Tey will be used without mention.
Lemma 2 (Lemma 14, see [7]).A model A of an h-inductive theory is pc if and only if for every positive formula φ and a∈ A, if A ⊭ φ(a), then there exists a positive formula ψ such that T ⊢ ¬∃x(φ(x) ∧ ψ(x)) and A ⊨ ψ(a).Lemma 3 (Theorem 2, see [7]).Every model on an hinductive theory T is continued in a pc model of T.
For every positive formula φ, we denote by Ctr T (φ) the set of positive formulas ψ such that T ⊢ ¬∃x(φ(x) ∧ ψ(x)). ( Two h-inductive theories are said to be companion if every model of one of them can be continued into a model of the other or equivalently if the theories have the same pc models. Every h-inductive theory T has a maximal companion denoted by T k (T) called the Kaiser's hull of T. T k (T) is the set of h-inductive sentences satisfed in each pc models of T. Likewise, T has a minimal companion denoted by T u (T), formed by its h-universal consequence sentences.Remark 4. Let T 1 and T 2 two h-inductive theories.Te following propositions are equivalent: Defnition 5. Let T be an h-inductive theory.
(i) T is said to be model complete if every model of T is a pc model of T. (ii) We say that T has a model companion whenever T k (T) is a model-complete theory.
Let A be a model of T. We shall use the following notations: (i) Diag + (A), the set of positive quantifer-free sentences satisfed by A in the language L(A).(ii) Diag(A), the set of atomic and negated atomic sentences satisfed by A in the language L(A).(ii) We denote by Defnition 6.Let A and B be two L-structures and h a homomorphism from A into B. h is said to be a strong immersion (in short s-immersion) if h is an immersion and B is a model of T i (A).

International Journal of Mathematics and Mathematical Sciences
Remark 7. Let A and B two L-structures.We have the following properties: (1 ) If A and B are two pc models of T, then every homomorphism from A into B is a strong immersion.Indeed, let φ(a, x) and ψ(a, x) be two positive formulas and let χ the h-inductive sentence ∀x(φ(a, x) ⟶ ψ(a, x)).Suppose that A ⊢ χ and B⊬χ, then there is b∈ B such that (5) Te pc models of the L(A)-theory T + (A) are the pc models of T that are the continuation of A. Indeed, it is clear that every pc model of T in which A is continued is a model of T + (A) and then a pc model of T + (A).Conversely, let B be a pc model Let A and B two be L-structures and f a mapping from A into B. We will use the following notations: Remark 8. Let A and B be two L-structures and f a mapping from A into B. Consider B as a L(A)-structure by interpreting the elements of A in B by f.We have the following:

Positive Amalgamation.
To abbreviate the nominations of homomorphism, embedding, immersion, and strong immersion in the defnition of the notions of amalgamation, we will use the frst letter of each mapping defned above.Defnition 9. Let Γ be a class of L-structures and A a member of Γ.We say that A is [h, e, i, s]-amalgamation basis of Γ, if for every B, C members of Γ, if A is continued into B by f and embedded into C by g, there exist D ∈ Γ, g ′ ∈ Imm(C, D), and f ′ ∈ Sm(B, D) such that the following diagram commutes: By the same way, we defne the notion of We We say that A is [α, c]-asymmetric amalgamation basis of Γ, whenever A is an [α, β, α, β]-amalgamation basis of Γ.
Te following remark lists some properties of diver forms of amalgamation with the notations and terminology given in the defnition above.

Remark 10 (1) Every L-structure A is an [i, h, s, h]-amalgamation
basis in the class of L-structures (lemma 4, [6]).Since every strong immersion is an immersion, it follows that every L-structure A is an [s, h]-asymmetric amalgamation basis in the class of L-structures.(2) Every L-structure A is an [s, i]-asymmetric amalgamation basis in the class of L-structures (lemma 5, [6]).(3) Every L-structure A is an [e, s]-asymmetric amalgamation basis in the class of L-structures (lemma 4, [9]).(4) Every L-structure A is an [i, h]-asymmetric amalgamation basis in the class of L-structures (lemma 8, [7]).( 5

) Every pc model of an h-inductive theory T is an
[h]-amalgamation basis in the class of model of T.
Lemma 11.Every L-structure is [s, x]-asymmetric amalgamation basis in the class of L-structure, where x is a homomorphism, an embedding, or an immersion.
Proof.Te proof of the lemma directly follows from the Remark 10.More precisely, the cases where x is a homomorphism is addressed in bullet 1 of the Remark 10, while the case where x is an embedding is covered in bullet 3. Te opposite direction follows easily from Lemma 12. □

Almost Positively Closed Structures
In this section, we introduce the notions of almost and Δ-almost positively closed models, and we give a syntactic characterization and a characterization via the closed formulas which turns out to be an essential tool in the study of the notion of Δ-almost positively closedness.
Defnition 14.Let T be an h-inductive theory and A a model of T. Let Δ be a subset of L-quantifer-free positive formulas such that for every φ(x) ∈ Δ, the set Proof.Assume that A is an Δ-apc model of T and let φ(a, x) ∈ Δ such that is inconsistent.Tus, there are a ′ , a 1 , a 2 , . . ., a n ∈ A and ψ(a, a ′ , a 1 , a 2 , . . ., a n ) ∈ Diag + (A) such that T ∪ ψ a, a′, a 1 , a 2 , . . ., a n , ¬ ∨ n i�1 φ a, a i  , ∃xφ(a, x)  , (5) is inconsistent, which implies For the other direction, let A be a model of T that satisfes the hypothesis of the theorem.Let φ(a, x) ∈ Δ and f ∈ Hom(A, B), where B is a model of T and B ⊨ ∃xφ(a, x).Given that A ⊨ ψ(a, a ′ , a 1 , . . ., a n ), then B ⊨ ψ(a, a ′ , a 1 , . . ., a n ) ∧ ∃xφ(a, x).By the hypothesis of the theorem, we obtain B ⊨ ∨ n i�1 φ(a, a i ).So, A is an Δ-apc model of T. On the other hand, since A is immersed in B, then there are a ′ , a 1 , . . ., a n ∈ A such that ψ(a, a ′ , a 1 , . . ., a n ) ∈ Diag + (A).By Teorem 15, A is an Δ-apc of T.

□
Lemma 17.Let (A i , f ij ) i≤j∈I be an inductive sequence of models of an h-inductive theory T. Suppose that for every i ∈ I, the model A i is Δ i -apc where Δ i is a set of quantifer-free positive L(A i )-formulas such that ∀i ≤ j ∈ I, Δ i ⊆ Δ j .Ten, the inductive limit A of the sequence

□
Remark 18.Let T be an h-inductive L-theory and Δ a set of quantifer-free positive L-formulas.We have the following properties: (1) If A is apc, then A is wpc of T. Example 1 (1) Let L � f   be a functional language.Let T be the hinductive theory.
Te theory T has a model companion axiomatized by Te class of apc model of T is elementary and axiomatized by the h-inductive theory.
(2) Let L and T be the functional language and the theory defned in the bullet above.Let T ″ the hinductive theory T ∪ ¬∃x(f(x) � x)  .Te class of apc model of T ″ is axiomatized by the h-inductive theory.
(3) Let T f the theory of felds.Since the negation of equality x � y is defned by the positive formula ∃z(x − y) • z � 1 and every homomorphism is an embedding then the classes of apc felds, pc felds, and existentially closed felds are equals.
Defnition 19.Let T be an h-inductive L-theory and A a model of T.
(i) A positive formula φ(x) is said to be T-algebraic if φ(x) ≢ ⊥ modulo T (i.e., φ(x) has a realisation in some model of T ) and there exists a positive formula ψ(y 1 , . . ., y n ) such that φ(x) ∧ ψ (y 1 , . . ., y n ) ≢ ⊥ modulo T + (A), and We denote by Al T the set of T-algebraic quantiferfree positive L-formulas.(ii) For every positive formula φ(x), we denote by E(φ, T) the set of positive formulas ψ(y) such that φ(x) ∧ ψ(y) ≢ ⊥ modulo T and satisfy the following property: (iii) A positive formula φ(x, y) is said to be (A, T)-closed if φ ≢ ⊥ modulo T, and for every pc model continuation Remark 20 (1) A quantifer-free positive formula is T-algebraic if and only if its algebraic is in the sense of Robinson ([4]).(2) Given that the class of pc models of T + (A) coincides with the class of pc models of T that are continuation of A (bullet 5 of Remark 7), then a formula is (A, T)-closed if and only if it is (A, T + (A))-closed.

International Journal of Mathematics and Mathematical Sciences
We denote by E T the set of quantifer-free positive formulas φ(x) such that E(φ, T) ≠ ∅.

Lemma 21. Let A be an h-amalgamation basis of T. If A is E T + (A) -wpc (resp. E T + (A) -apc), then every formula in Al T
Proof.Let A be a E T + (A) -wpc and an h-amalgamation basis of T. Assume the existence of a formula φ(a, y) ∈ Al T + (A) such that φ(a, y) is not (A, T)-closed.So, there exist a pc models ′ , a contradiction.Te proof of the case where A is Al T -apc is an immediate.

Strong Amalgamation
In this section, we introduce the notions of positive strong amalgamation and h-strong amalgamation.We investigate their properties and interactions with the notion of almost positively closedness.

Positive Strong Amalgamation
Defnition 22.Let T be an h-inductive theory.A model A of T is said to be a positive strong amalgamation short PSA) (resp.h-strong amalgamation basis (in short h-SA)) of T, if for every pc models (resp.models) B and C of T, if A is continued into B and C by two homomorphisms f and g, respectively, then there exist D a model of T and f ′ , g ′ , two homomorphisms, such that the following diagram commutes: and satisfes the following property (P): Remark 23.Note that in the defnition of PSA basis, we can reformulate the property (P) as follows: , then there exist a, a ′ ∈ A such that c � g(a) and b � f(a ′ ).
Indeed, let a, a ′ ∈ A such that c � g(a) and b � f(a ′ ), then Given that g ′ is an immersion, we have Example 2 (1) Every h-inductive theory for which the unique pc model is the trivial model A e � a { } has the positive strong amalgamation property.As examples of these theories, we have the theory of groups and the theory of partially ordered sets.
(2) Let L � p, q   where p and q are two unary relation symbols.Let T be the h-inductive theory ∀x, y((p(x) ∧ q(y)) Proof.Te proof consists in the verifcation that the following set is L(B ∪ C)-consistent.
where every elements of A is interpreted by the same symbols of constant in B and C. Given that B ⊨ ψ(a, b) and A is immersed in B, so there is a Let D be a model of T ′ , then the following digram commutes: where h ′ is a homomorphism and s a strong immersion.Let b ∈ B and c ∈ C such that s(c) � h ′ (b), so there is a, a ′ in A such that c � h(a) and b � i(a ′ ).By the commutativity of the diagram above, we have Given that s is an immersion, we obtain

□ Corollary 25. Every pc model A of T is a h-strong amalgamation basis of T.
Proof.Immediate from Lemma 24.

□ Proposition 2 . Let A and B be two models of an h-inductive theory T. If
Proof.Let A 1 and A 2 be two models of T, f ∈ Hom(A, A 1 ), and g ∈ Hom(A, A 2 ).By applying Lemma 24 to the diagrams A 1 ← A ⟶ B and A 1 ← A ⟶ B, we get the commutative diagrams (1) and ( 2), where f ′ , g ′ are homomorphisms and i 1 , i 2 are strong immersions.Now, given that B has the h-strong amalgamation property, we get the commutative diagram (3): where f ″ , g ″ are homomorphisms.
We claim that C makes the diagram A 1 ← A ⟶ A 2 strongly amalgamate.Indeed, let a 1 ∈ A 1 and a 2 ∈ A 2 such that f ″ °i1 (a 1 ) � g ″ °i2 (a 2 ).By the h-strong amalgamation property of the diagram (3), there is b ∈ B such that f ′ (b) � i 1 (a 1 ) and g ′ (b) � i 2 (a 2 ).Considering the properties of the diagrams (1) and (2), we get two elements a and a ′ from A such that Given that i is an immersion, then a � a ′ and f(a) � a 1 , g(a) � a 2 .So, A is a h-SA basis of T.

□ Lemma 27. An h-amalgamation basis A of T is a PSA basis if and only if for every pc model of T + (A) and for every φ(a, x) ∈ E T
Proof.Let A be a PSA basis of T. Suppose that there are φ(a, x) ∈ E T + (A) and B a pc model of T Given that A is a PSA basis of T, we obtain the following commutative diagram.(25) Now, since D ⊨ φ(a, i(b)) ∧ ψ(a, i ′ (c)), then D ⊨ ∨ i,j i(b i ) � i ′ (c j ), which implies the existence of an element a ′ ∈ A such that i(b i ) � i ′ (c j ) � i(a ′ ), a contradiction.
For the other direction, let B and C be two pc models of T, f ∈ Hom(A, B), and g ∈ Hom(A, C) such that the following diagram is not h-strongly amalgamable.□ (i) Hom(A, B): the set homomorphisms from A into B. (ii) Emb(A, B): the set embeddings from A into B. (iii) Imm(A, B): the set immersions from A into B. (iv) Sm(A, B): the set s-immersions from A into B.

( 2 )( 4 )
Every pc model of T is an apc (resp.Δ-apc) model of T. (3) Te classes of apc and wpc (resp.Δ-apc and Δ-wpc) models of T are h-inductive.If A is an apc model of T and B a model of T, then Emb(A, B) � Imm(A, B). (5) Let Δ ⊆ D be two sets of free quantifer positive formulas.If A is D-apc (resp.D-wpc) then A is Δ-apc (resp.Δ-wpc).(6) Every apc model of T has the property of [e, h]-asymmetric amalgamation (property 4 of Remark 18, and the property 4 of Remark 10).

( 3 )
Let A be a model of T. Denote by C A the set of quantifer-free formulas that are (A, T)-closed.Ten, A is C A -wpc. (4) If every formula in Al T + (A) is (A, T)-closed, by the bullet 2 above, the model A is Al T + (A) -wpc, and since Al T ⊂ Al T + (A) , A is also Al T -wpc.
is the unique pc model of T; thus, T has the positive strong amalgamation property.However, the structure A � a { } where A ⊭ (p(a)∨q(a)) has no h-strong amalgamation property.Indeed, let B � a, b { }, C � a, c { }, B ⊨ p(b), and C ⊨ q(c), then the diagram B ← A ⟶ C cannot be h-strongly amalgamate.Lemma 24.Let A, B, and C be three L-structures.Let i ∈ Imm(A, B) and h ∈ Hom(A, C).Ten, there exist D a Lstructure, h ′ a homomorphism, and s an s-immersion such that the following diagram commutes: and satisfes the following property: ∀(b, c) ∈ B × C, and if h ′ (b) � s(c), then there exists a ∈ A such that c � h(a) and b � i(a).

( 1 )where i 1
exist φ(f(a), b 1 • • • b n ) ∈ Diag + (B), ψ(g(a), c 1 • • • c m ) ∈ Diag + (C) where b 1 , . . ., b n ∈ B − A and c 1 , . . ., c m ∈ C − A such that T + (A) ⊢ ∀y (φ(a, x) ∧ ψ(a, y)) ⟶ ∨ i,j x i � y j  .(27) Ten, φ(a, x) ∈ E T + (A) and B ⊨ φ(a, b 1 • • • b n ).□Theorem 28.Let A be an h-amalgamation basis of T, then we have the following properties:(1) If A is a E T + (A) -wpc model of T, then A is a PSA basis of T.International Journal of Mathematics and Mathematical Sciences(2) If A is a PSA basis of T, then A is Al T + (A) -wpc.Proof Let A be an h-amalgamation basis and a E T + (A) -wpc model of T. Let B and C be two pc models of T, f ∈ Hom(A, B), and g ∈ Hom(A, C).Let D a model of T such that the following diagram commutes: and i 2 are immersions.We claim that the setT ∪ Diag + (B) ∪ Diag + (C) ∪ b ≠ c | b ∈ B − A, c ∈ C − A { } is L(B ∪ C)-consistent (notethat the element of A are interpreted by the same symbols of constants in B and C).Assume that the set above is inconsistent.Ten there are a∈ A, b∈ B − A, c∈ C − A, φ(a, b) ∈ Diag + (B) and ψ(a, c) ∈ Diag + (C) such thatT + (A) ∪ φ(a, b), ψ(a, c), ∧ i,j b i ≠ c j  ,(29)is L(B ∪ C)-inconsistent, thereby T + (A) ⊢ ∀y, z (φ(a, y) ∧ ψ(a, z)) ⟶ ∨ C ⊨ ψ(a, c), ψ ∈ E T + (A), and A is anE T + (A) -wpc model, then there is a ′ ∈ A such that C ⊨ ψ(a, a ′ ).Tereby, D ⊨ ψ(a, a ′ ), so B ⊨ ψ(a, a ′ ) ∧ φ(a, b), which implies B ⊨ ∨ i,j b i � a j ′ , a contradiction.Tus, A is a PSA basis of T. (2) Suppose that A is PSA of T. Since Al T + (A) ⊆ E T + (A) , byLemma 27, every formula in Al T + (A) is (A, T)-closed, which implies that A is a Al T + (A) -wpc model of T by Remark 20 bullet (4).
International Journal of Mathematics and Mathematical SciencesOtherwise, we can fnd two continuations of A in which one of them satisfes φ(a) and the other does not satisfy φ(a), which contradicts the assumption that A has the [h, i]-symmetric amalgamation property.An h-inductive theory T has a model companion if and only if T k (T) has the [h, i]symmetric amalgamation property.Proof.Suppose that T has a model companion, then every model of T k (T) is a pc model.Since the pc models have the [h]-amalgamation property and the homomorphisms between the pc models are immersion, it follows from the ffth bullet of the Remark 10 that T k (T) has the [h, i]-symmetric amalgamation property.
Tecase where x is an immersion is addressed in bullet 2. □Lemma 12.A model of T is pc if and only if it has the [h, i]-symmetric amalgamation property in the class of models of T.Proof.Let A be an [h, i]-symmetric amalgamation basis of T. Assume that A ⊭ φ(a), where a∈ A and φ a positive formula.Given that A is an [h, i]-symmetric amalgamation basis, we claim that T ∪ Diag + (A) ∪ φ(a)   is inconsistent.□Proposition13.
positively closed (apc in short), if for every model B ⊨ T, f ∈ Hom(A, B), and φ(x, y) a quantifer-free positive formula, if B ⊨ ∃yφ(a, y) and a∈ A, then there is a ′ ∈ A such that B ⊨ φ(a, a ′ ).(ii) Δ-almost positively closed (Δ-apc in short), if for every model B ⊨ T, f ∈ Hom(A, B), and φ(x, y) ∈ Δ, if B ⊨ ∃yφ(a, y) and a∈ A, then there is a ′ ∈ A such that B ⊨ φ(a, a ′ ).(iii) Weakly almost positively closed (wpc in short), if for every pc model B ⊨ T, f ∈ Hom(A, B), and φ(x, y) a quantifer-free positive formula if B ⊨ ∃yφ (a, y) and a∈ A, then there is a ′ ∈ A such that B ⊨ φ(a, a ′ ).(iv) Δ-weakly almost positively closed (Δ-wpc in short), if for every pc model B ⊨ T, f ∈ Hom(A, B), and φ(x, y) ∈ Δ, if B ⊨ ∃xφ(a, x), then there is a ′ ∈ A such that B ⊨ φ(a, a ′ ).
15. Let A be a model of an h-inductive L-theory T.Let Δ be a set of L(A)-quantifer free positive formula that satisfes the condition of Defnition 14. Te model A is Δ-apc of T if and only if for every φ(a, x) ∈ Δ, there exist n ∈ N and a quantifer-free positive formula ψ(a, a ′ , a 1 , . . ., a n ) ∈ Diag + (A) such that T ⊢ ∀x, y, y 1 , . . ., y n ψ x, y, y 1 , . . ., y n  ∧ ∃zφ(x, z)  ⟶ ∨ n i�1 φ x, y i  .