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*MML* *Identifier*: REALSET2.

"... The articles [4], [2], [3], and [1] provide the notation and terminology for thispaper. A field structure is said to be a field if: (Def.1) there exists an at least 2-elements set A and there exists a binary op-eration o1 of A and there exists an element n1 of A and there exists abinary operation o2 ..."

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The articles [4], [2], [3], and [1] provide the notation and terminology for thispaper. A field structure is said to be a field if: (Def.1) there exists an at least 2-elements set A and there exists a binary op-eration o1 of A and there exists an element n1 of A and there exists abinary operation o2 of A preserving A \ {n1} and there exists an element n2 of A \ single(n1) such that it = field(A, o1, o2, n1, n2) and group(A, o1, n1) is a group and for every non-empty set B and for every binary oper-ation P of B and for every element e of B such that B = A \ single(n1)and e = n2 and P = o2 _n1 A holds group(B, P, e) is a group and for allelements

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*MML* *Identifier*:BVFUNC_9.

"... Summary. In this paper, we have proved some elementary propositional calculus formulae ..."

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Summary. In this paper, we have proved some elementary propositional calculus formulae

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*MML* *Identifier*:RMOD_4.

"... provide the notation and terminology for this paper. For simplicity, we adopt the following rules: R denotes a ring, V denotes a right module over ..."

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provide the notation and terminology for this paper. For simplicity, we adopt the following rules: R denotes a ring, V denotes a right module over

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*MML* *Identifier*: TOPRNS 1.

"... notation and terminology for this paper. For simplicity we adopt the following rules: f denotes a function, N, n, m denote natural numbers, q, r, r1, r2 denote real numbers, x is arbitrary, and w, w1, w2, g denote points of EN T. Let us consider N. A sequence in E N T E N T. In the sequel s1, s2, s3 ..."

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notation and terminology for this paper. For simplicity we adopt the following rules: f denotes a function, N, n, m denote natural numbers, q, r, r1, r2 denote real numbers, x is arbitrary, and w, w1, w2, g denote points of EN T. Let us consider N. A sequence in E N T E N T. In the sequel s1, s2, s3, s4, s ′ 1 are sequences in EN T. is a function from ¦ into the carrier of Next we state two propositions: (1) f is a sequence in EN T if and only if domf ¦ = and for every x such that x ¦ ∈ holds f(x) is a point of EN T. (2) f is a sequence in EN T iff dom f ¦ = and for every n holds f(n) is a point of EN T. Let us consider N, s1, n. Then s1(n) is a point of EN T. Let us consider N. A sequence in EN T is non-zero if: (Def.1) rng it ⊆ (the carrier of EN T) \ {0EN T We now state several propositions: (3) s1 is non-zero iff for every x such that x ¦ ∈ holds s1(x) � = 0EN. T (4) s1 is non-zero iff for every n holds s1(n) � = 0 E N T (5) For all N, s1, s2 such that for every x such that x ¦ ∈ holds s1(x) = s2(x) holds s1 = s2. (6) For all N, s1, s2 such that for every n holds s1(n) = s2(n) holds s1 = s2. (7) For every point w of EN T there exists s1 such that rng s1 = {w}. The scheme ExTopRealNSeq deals with a natural number A and a unary functor F yielding a point of E A T, and states that:

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Some Properties of the Intervals *MML* *Identifier*:MEASURE6.

"... for this paper. The following propositions are true: (1) There exists a function F from N into [:N, N:] such that F is one-to-one and domF = N and rngF = [:N, N:]. (2) For every function F from N into R such that F is non-negative holds 0 R ≤ ∑F. (3) Let F be a function from N into R and x be an ext ..."

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for this paper. The following propositions are true: (1) There exists a function F from N into [:N, N:] such that F is one-to-one and domF = N and rngF = [:N, N:]. (2) For every function F from N into R such that F is non-negative holds 0 R ≤ ∑F. (3) Let F be a function from N into R and x be an extended real number. Suppose there exists a natural number n such that x ≤ F(n) and F is non-negative. Then x ≤ ∑F. (8) 1 For all extended real numbers x, y such that x is a real number holds (y − x)+x = y and (y+x) − x = y. (10) 2 For all extended real numbers x, y, z such that z ∈ R and y < x holds (z+x)−(z+y) = x−y. (11) For all extended real numbers x, y, z such that z ∈ R and x ≤ y holds z + x ≤ z + y and x+z ≤ y+z and x − z ≤ y − z. (12) For all extended real numbers x, y, z such that z ∈ R and x < y holds z + x < z + y and x+z < y+z and x − z < y − z. Let x be a real number. The functor R(x) yielding an extended real number is defined as follows: (Def. 1) R(x) = x. We now state a number of propositions: (13) For all real numbers x, y holds x ≤ y iff R(x) ≤ R(y). (14) For all real numbers x, y holds x < y iff R(x) < R(y). (15) For all extended real numbers x, y, z such that x < y and y < z holds y is a real number. (16) Let x, y, z be extended real numbers. Suppose x is a real number and z is a real number and x ≤ y and y ≤ z. Then y is a real number. (17) For all extended real numbers x, y, z such that x is a real number and x ≤ y and y < z holds y is a real number.

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*MML* *Identifier*:CATALG_1. Algebra of Morphisms

"... Let I be a set and let A, f be functions. The functor f ↾ IA yields a many sorted function indexed by I and is defined as follows: (Def. 1) For every set i such that i ∈ I holds ( f ↾ IA)(i) = f↾A(i). One can prove the following propositions: (1) For every set I and for every many sorted set A inde ..."

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Let I be a set and let A, f be functions. The functor f ↾ IA yields a many sorted function indexed by I and is defined as follows: (Def. 1) For every set i such that i ∈ I holds ( f ↾ IA)(i) = f↾A(i). One can prove the following propositions: (1) For every set I and for every many sorted set A indexed by I holds id � A ↾ IA = idA. (2) Let I be a set, A, B be many sorted sets indexed by I, and f, g be functions. If rng κ ( f ↾ IA)(κ) ⊆ B, then (g · f) ↾ IA = (g ↾ IB) ◦(f ↾ IA). (3) Let f be a function, I be a set, and A, B be many sorted sets indexed by I. Suppose that for every set i such that i ∈ I holds A(i) ⊆ dom f and f ◦ A(i) ⊆ B(i). Then f ↾ IA is a many sorted function from A into B. (4) Let A be a set, i be a natural number, and p be a finite sequence. Then p ∈ A i if and only if len p = i and rng p ⊆ A. (5) Let A be a set, i be a natural number, and p be a finite sequence of elements of A. Then p ∈ A i if and only if len p = i. (6) For every set A and for every natural number i holds A i ⊆ A ∗.

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Trigonometric Form of Complex Numbers *MML* *Identifier*: COMPTRIG.

"... The scheme Regr without 0 concerns a unary predicate P, and states that: P [1] provided the parameters meet the following requirements: • There exists a non empty natural number k such that P [k], and • For every non empty natural number k such that k � = 1 and P [k] there exists a non empty natural ..."

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The scheme Regr without 0 concerns a unary predicate P, and states that: P [1] provided the parameters meet the following requirements: • There exists a non empty natural number k such that P [k], and • For every non empty natural number k such that k � = 1 and P [k] there exists a non empty natural number n such that n < k and P [n]. The following propositions are true: (3) 1 For every element z of C holds ℜ(z) ≥ −|z|. (4) For every element z of C holds ℑ(z) ≥ −|z|. (5) For every element z of CF holds ℜ(z) ≥ −|z|. (6) For every element z of CF holds ℑ(z) ≥ −|z|. (7) For every element z of CF holds |z | 2 = ℜ(z) 2 + ℑ(z) 2. (8) For all real numbers x1, x2, y1, y2 such that x1 + x2iCF = y1 + y2iCF holds x1 = y1 and x2 = y2. (9) For every element z of CF holds z = ℜ(z) + ℑ(z)iCF. (10) 0CF = 0 + 0iCF. (12) 2 For every unital non empty groupoid L and for every element x of L holds power L (x, 1) = x. (13) For every unital non empty groupoid L and for every element x of L holds power L (x, 2) = x · x. (14) Let L be an add-associative right zeroed right complementable right distributive unital non empty double loop structure and n be a natural number. If n> 0, then power L (0L, n) = 0L. 1 The propositions (1) and (2) have been removed. 2 The proposition (11) has been removed.

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Some Properties of the Intervals *MML* *Identifier*: MEASURE6.

"... and notation for this paper. The scheme FunctXD YD concerns a non empty set A, a non empty set B, and a binary predicate P, and states that: There exists a function F from A into B such that for every element x of A holds P[x,F(x)] provided the following condition is satisfied: • For every element x ..."

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and notation for this paper. The scheme FunctXD YD concerns a non empty set A, a non empty set B, and a binary predicate P, and states that: There exists a function F from A into B such that for every element x of A holds P[x,F(x)] provided the following condition is satisfied: • For every element x of A there exists an element y of B such that P[x,y]. Let X, Y be non empty sets. Note that Y X is non empty. We now state a number of propositions: (1) There exists a function F from ¦ into [: ¦ , ¦:] such that F is one-to-one and domF = ¦ and rng F = [: ¦ , ¦:]. (2) For every function F from ¦ into ¤ such that F is non-negative holds 0 § ≤ � F. (3) Let F be a function from ¦ into ¤ and let x be a Real number. Suppose there exists a natural number n such that x ≤ F(n) and F is non-negative. Then x ≤ � F. (4) For every Real number x such that there exists a Real number y such that y < x holds x � = −∞. (5) For every Real number x such that there exists a Real number y such that x < y holds x � = +∞. (6) For all Real numbers x, y holds x ≤ y iff x < y or x = y. (7) Let x, y be Real numbers and let p, q be real numbers. If x = p and y = q, then p ≤ q iff x ≤ y. (8) For all Real numbers x, y such that x is a real number holds (y−x)+x = y and (y + x) − x = y. (9) For all Real numbers x, y such that x ∈ ¤ holds x + y = y + x.

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*MML* *Identifier*: WAYBEL18. Injective Spaces 1

"... terminology for this paper. One can prove the following propositions: 1. PRODUCT TOPOLOGIES (1) Let x, y, z, Z be sets. Then Z ⊆ {x,y,z} if and only if one of the following conditions is satisfied: ..."

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terminology for this paper. One can prove the following propositions: 1. PRODUCT TOPOLOGIES (1) Let x, y, z, Z be sets. Then Z ⊆ {x,y,z} if and only if one of the following conditions is satisfied:

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*MML* *Identifier*: YONEDA_1. Yoneda Embedding

"... for this paper. In this paper A is a category, a is an object of A, and f is a morphism of A. Let us consider A. The functor EnsHomA yields a category and is defined by: (Def. 1) EnsHomA = Ens Hom(A). The following propositions are true: (1) Let f, g be functions and m1, m2 be morphisms of EnsHomA. ..."

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for this paper. In this paper A is a category, a is an object of A, and f is a morphism of A. Let us consider A. The functor EnsHomA yields a category and is defined by: (Def. 1) EnsHomA = Ens Hom(A). The following propositions are true: (1) Let f, g be functions and m1, m2 be morphisms of EnsHomA. If codm1 = domm2 and 〈〈domm1, codm1〉, f 〉 = m1 and 〈〈domm2, codm2〉, g 〉 = m2, then 〈〈domm1, codm2〉, g · f 〉 = m2 · m1. (2) hom(a,−) is a functor from A to EnsHomA. Let us consider A, a. The functor hom F (a,−) yields a functor from A to EnsHomA and is defined as follows: (Def. 2) hom F (a,−) = hom(a,−). The following proposition is true (3) For every morphism f of A holds hom F (cod f,−) is naturally transformable to hom F (dom f,−).