hcju/proofwikips huggingface dataset:
Injection Induces Well-Ordering
Let $\left({T, \le}\right)$ be a well-ordered set. Let $S$ be a set. Let $f: S \to T$ be an injection. Then $\le_f$, the ordering induced by $f$, is a well-ordering.
Hartogs' Lemma (Set Theory)
Let $S$ be a set. Then there exists an ordinal $\alpha$ such that there is no injection from $\alpha$ to $S$.
Bourbaki-Witt Fixed Point Theorem
Let $\struct {X, \le}$ be a non-empty chain complete poset (that is, an ordered set in which every chain has a supremum). Let $f: X \to X$ be an inflationary mapping, that is, so that $\map f x \ge x$. Then for every $x \in X$ there exists $y \in X$ where $y \ge x$ such that $\map f y = y$.
Convergent Sequence with Finite Number of Terms Deleted is Convergent
Let $\left({X, d}\right)$ be a metric space. Let $\left\langle{x_k}\right\rangle$ be a sequence in $X$. Let $\left\langle{x_k}\right\rangle$ be convergent. Let a finite number of terms be deleted from $\left \langle {x_k} \right \rangle$. Then the resulting subsequence is convergent.
Cauchy Condensation Test
Let $\sequence {a_n}: n \mapsto \map a n$ be a decreasing sequence of strictly positive terms in $\R$ which converges with a limit of zero. That is, for every $n$ in the domain of $\sequence {a_n}$: $a_n > 0$, $a_n \ge a_{n + 1}$, and $a_n \to 0$ as $n \to +\infty$. Then the series: :$\displaystyle \sum_{n \mathop = 1}^\infty a_n$  converges  the condensed series: :$\displaystyle \sum_{n \mathop = 1}^\infty 2^n \map a {2^n}$ converges.
Between Two Sets Exists Injection or Surjection
Let $S$ and $T$ be sets. Then either or both of the following cases hold: :$(1):$ There exists a mapping $f: S \to T$ such that $f$ is an injection :$(2):$ There exists a mapping $f: S \to T$ such that $f$ is a surjection.
Injection from Set to Power Set
For every set $S$, there exists an injection from $S$ to its power set $\mathcal P \left({S}\right)$.
Minimum Degree is at Least Connectivity
Let $G = \left({V, E}\right)$ be a simple graph. Then: :$\delta \left({G}\right) \geq \kappa \left({G}\right)$ That is, the minimum degree of $G$ is at least its connectivity.
Homeomorphism Relation is Equivalence
Let $T_1$ and $T_2$ be topological spaces. Let $T_1 \sim T_2$ denote that $T_1$ and $T_2$ are homeomorphic. The relation $\sim$ is an equivalence relation.
Identity Mapping is Homeomorphism
Let $T$ be a topological space. The identity mapping $I_T: T \to T$ defined as: :$\forall x \in T: I_T \left({x}\right) = x$ is a homeomorphism.
Composite of Homeomorphisms is Homeomorphism
Let $T_1, T_2, T_3$ be topological spaces.  Let $f: T_1 \to T_2$ and $g: T_2 \to T_3$ be homeomorphisms. Then $g \circ f: T_1 \to T_3$ is also a homeomorphism.
Identity Mapping is Continuous
Let $T = \struct {S, \tau}$ be a topological space. The identity mapping $I_S: S \to S$ defined as: :$\forall x \in S: \map {I_S} x = x$ is a continuous mapping.
Kuratowski's Lemma
Let $\struct {S, \preceq}, S \ne \O$ be a non-empty ordered set.  Then every chain in $S$ is the subset of some maximal chain.
Included Set Topology is Topology
Let $T = \left({S, \tau_H}\right)$ be an included set space. Then $\tau_H$ is a topology on $S$, and $T$ is a topological space.
Space of Compact Linear Transformations is Banach Space
Let $H, K$ be Hilbert spaces, and let $B_0 \left({H, K}\right)$ be the space of compact linear transformations from $H$ to $K$. Let $\Bbb F \in \left\{{\R, \C}\right\}$ be the ground field of $K$. Now $B_0 \left({H, K}\right) \subseteq K^H$, the set of mappings from $H$ to $K$. Therefore, $B_0 \left({H, K}\right)$ can be endowed with pointwise addition ($+$) and ($\Bbb F$)-scalar multiplication ($\circ$). Let $\left\Vert{\cdot}\right\Vert$ denote the norm on bounded linear operators. Then $\left\Vert{\cdot}\right\Vert$ is a norm on $B_0 \left({H, K}\right)$. Furthermore, $B_0 \left({H, K}\right)$ is a Banach space with respect to this norm.
Compact Linear Transformation is Bounded
Let $H, K$ be Hilbert spaces. Let $T \in B_0 \left({H, K}\right)$ be a compact linear transformation. Then $T$ is also a bounded linear transformation. That is, $B_0 \left({H, K}\right) \subseteq B \left({H, K}\right)$.
Compact Linear Transformations Composed with Bounded Linear Operator
Let $H, K$ be Hilbert spaces. Let $T \in B_0 \left({H, K}\right)$ be a compact linear transformation. Let $A \in B \left({H}\right), B \in B \left({K}\right)$ be bounded linear operators. Then the compositions $TA$ and $BT$ are also compact linear transformations.
Linear Transformation Compact iff Adjoint Compact
Let $H, K$ be Hilbert spaces. Let $T: H \to K$ be a linear transformation. Then $T$ is compact iff its adjoint $T^*$ is.
Finite Rank Operators Dense in Compact Linear Transformations
Let $H, K$ be Hilbert spaces. Then: :$\map {B_{00} } {H, K}$ is everywhere dense in $\map {B_0} {H, K}$ where: :$\map {B_{00} } {H, K}$ is the space of continuous finite rank operators from $H$ to $K$ :$\map {B_0} {H, K}$ is the space of compact linear transformations from $H$ to $K$. That is, for every $T \in \map {B_0} {H, K}$, there is a sequence $\sequence {T_n}_{n \mathop \in \N}$ in $\map {B_{00} } {H, K}$ such that: :$\ds \lim_{n \mathop \to \infty} \norm {T_n - T} = 0$ where $\norm {\, \cdot \,}$ denotes the norm on bounded linear transformations.
Inverse of Matrix Product
Let $\mathbf {A, B}$ be square matrices of order $n$ Let $\mathbf I$ be the $n \times n$ unit matrix. Let $\mathbf A$ and $\mathbf B$ be invertible. Then the matrix product $\mathbf {AB}$ is also invertible, and: :$\paren {\mathbf A \mathbf B}^{-1} = \mathbf B^{-1} \mathbf A^{-1}$
Closure of Range of Compact Linear Transformation is Separable
Let $H, K$ be Hilbert spaces. Let $T \in \map {B_0} {H, K}$ be a compact linear transformation. Then $\map \cl {\Rng T}$ is separable.
Elementary Row Operations as Matrix Multiplications
Let $e$ be an elementary row operation. Let $\mathbf E$ be the elementary row matrix of order $m$ defined as: :$\mathbf E = e \paren {\mathbf I}$ where $\mathbf I$ is the unit matrix. Then for every $m \times n$ matrix $\mathbf A$: :$e \paren {\mathbf A} = \mathbf {E A}$ where $\mathbf {E A}$ denotes the conventional matrix product.
Included Set Topology on Finite Intersection
Let $T = \struct {S, \tau}$ be a topological space on a set $S$. Let $A_1, A_2, \ldots, A_n$ be a finite set of subsets of $S$: :$\forall i \in \closedint 1 n: A_i \subseteq S$ Let $\forall i \in \closedint 1 n: \map T {A_i} = \struct {S, \tau_{A_i} }$ be the included set spaces on $S$ by $A_i$. Let: :$\forall i \in \closedint 1 n: \map T {A_i} \le T$ where $\map T {A_i} \le T$ denotes that $\map T {A_i}$ is coarser than $T$. Then: :$\displaystyle \map T {\bigcap A_i} \le T$ where $\displaystyle \map T {\bigcap A_i}$ is the included set space on $S$ by $\displaystyle \bigcap_{i \mathop = 1}^n A_i$.
Included Set Topology on Union
Let $T = \struct {S, \tau}$ be a topological space on a set $S$. Let $\family {A_i}_{i \mathop \in I}$ be a family of subsets of $S$ indexed by the indexing set $I$: :$\forall i \in I: A_i \subseteq S$ Let $\forall i \in I: \map T {A_i} = \struct {S, \tau_{A_i} }$ be the included set spaces on $S$ by $A_i$. Let: :$\forall i \in I: \map T {A_i} \ge T$ where $\map T {A_i} \ge T$ denotes that $\map T {A_i}$ is finer than $T$. Then: :$\map T {\bigcup A_i} \ge T$ where $\map T {\bigcup A_i}$ is the included set space on $S$ by $\displaystyle \bigcup_{i \mathop \in I} A_i$.
Kuratowski's Free Set Theorem
Let $n \in \Z_{>0}$ be a positive integer. Let $X$ be an infinite set. Let $\left[{X}\right]^n$ denote the set of $n$-element subsets of $X$ whose cardinality is $n$. Let $\left[{X}\right]^{< \aleph_0}$ denote the set of finitesubsets of $X$. Then: : the cardinality of $X$ is greater than or equal to $\aleph_n$  : for every mapping $f: \left[{X}\right]^n \to \left[{X}\right]^{<\aleph_0}$ there exists an $(n + 1)$-element free subset of $X$ with respect to $f$. That is, a subset $Y$ of $X$ with $n+1$ elements such that $y \notin f \left({Y \setminus \left\{{y}\right\} }\right)$ for all $y \in Y$.
Diagonalizable Operator Bounded iff Value Set Bounded
Let $H$ be a Hilbert space. Let $A: H \to H$ be a diagonalizable operator. Let $\left({\alpha_e}\right)_{e \in E}$ be the value set of $A$, with respect to a suitable basis $E$ for $H$. Then $A$ is bounded iff $\left({\alpha_e}\right)_{e \in E}$ is, i.e., iff: :$\exists M \in \R: \forall e \in E: \left\vert{\alpha_e}\right\vert \le M$
Diagonalizable Operator Compact iff Value Set Converges to Zero
Let $H$ be a Hilbert space of countable dimension. Let $A: H \to H$ be a diagonalizable operator. Let $\left({\alpha_n}\right)_{n \in \N}$ be the value set of $A$, with respect to a suitable basis $E = \left({e_n}\right)_{n \in \N}$ for $H$. Then $A$ is compact iff: :$\displaystyle \lim_{n \to \infty} \alpha_n = 0$
Nonzero Eigenvalue of Compact Operator has Finite Dimensional Eigenspace
Let $H$ be a Hilbert space. Let $T \in B_0 \left({H}\right)$ be a compact operator. Let $\lambda \in \sigma_p \left({T}\right), \lambda \ne 0$ be a nonzero eigenvalue of $T$. Then the eigenspace for $\lambda$ has finite dimension.
Condition for Nonzero Eigenvalue of Compact Operator
Let $H$ be a Hilbert space over $\Bbb F \in \set {\R, \C}$. Let $T \in \map {B_0} H$ be a compact operator. Let $\lambda \in \Bbb F, \lambda \ne 0$ be a nonzero scalar. Suppose that the following holds: :$\inf \set {\norm {\paren {T - \lambda I} h}_H: \norm h_H = 1} = 0$ Then $\lambda \in \map {\sigma_p} T$, that is, $\lambda$ is an eigenvalue for $T$.
Finite Rank Operator is Compact
Let $H, K$ be Hilbert spaces. Let $T \in \map {B_{00} } {H, K}$ be a bounded finite rank operator. Then $T \in \map {B_0} {H, K}$, that is, $T$ is compact.
Adjoint of Finite Rank Operator
Let $H, K$ be Hilbert spaces. Let $T \in \map {B_{00} } {H, K}$ be a bounded finite rank operator. Then: :$T^* \in \map {B_{00} } {K, H}$ that is, the adjoint of $T$ is also a bounded finite rank operator.
Compact Idempotent is of Finite Rank
Let $H$ be a Hilbert space. Let $T \in \map {B_0} H$ be a compact linear operator, and let $T$ be idempotent. Then: :$T \in \map {B_{00} } H$ that is, $T$ is a bounded finite rank operator.
Characterization of Finite Rank Operators
Let $H$ be a Hilbert space. Let $T \in B_{00} \left({H}\right)$ be a bounded finite rank operator. Let $n = \operatorname{dim} \left({\operatorname{ran} T}\right)$ be the rank of $T$. Then there are orthonormal vectors $e_1, \ldots, e_n$ and vectors $g_1, \ldots, g_n$ of $H$ such that: :$\forall h \in H: Th = \displaystyle \sum_{i=1}^n \left\langle{h, e_i}\right\rangle_H g_i$
Compact Operator on Hilbert Space Direct Sum
Let $\sequence {H_n}_{n \mathop \in \N}$ be a sequence of Hilbert spaces. Denote by $H = \displaystyle \bigoplus_{n \mathop = 1}^\infty H_n$ their Hilbert space direct sum. For each $n \in \N$, let $T_n \in \map B {H_n}$ be a bounded linear operator. Suppose that one has $\displaystyle \sup_{n \mathop \in \N} \norm {T_n} < \infty$, where $\norm {\, \cdot \, }$ signifies the norm on bounded linear operators. Define $T \in \map B H$ by:  :$\forall h = \sequence {h_n}_{n \mathop \in \N}: T h = \sequence {T_n h_n}_{n \mathop \in \N} \in H$ (That $T$ is indeed bounded follows from Bounded Linear Operator on Hilbert Space Direct Sum.) Then $T$ is compact  the following conditions hold: : For each $n \in \N$, $T_n$ is compact : $\displaystyle \lim_{n \mathop \to \infty} \norm {T_n} = 0$
Bounded Linear Operator on Hilbert Space Direct Sum
Let $\left({H_n}\right)_{n \in \N}$ be a sequence of Hilbert spaces. Denote by $H = \displaystyle \bigoplus_{n \mathop = 1}^\infty H_n$ their Hilbert space direct sum. For each $n \in \N$, let $T_n \in B \left({H_n}\right)$ be a bounded linear operator. Suppose that one has $\displaystyle \sup_{n \mathop \in \N} \, \left\Vert{T_n}\right\Vert < \infty$, where $\left\Vert{\cdot}\right\Vert$ signifies the norm on bounded linear operators. Define $T: H \to H$ by:  :$\forall h = \left({h_n}\right)_{n \in \N}: T h = \left({T_n h_n}\right)_{n \in \N} \in H$ Then $T \in B \left({H}\right)$ is a bounded linear operator.
Compact Self-Adjoint Operator has Countable Point Spectrum
Let $H$ be a Hilbert space. Let $T \in B_0 \left({H}\right)$ be a compact, self-adjoint operator. Then its point spectrum $\sigma_p \left({T}\right)$ is countable.
Spectral Theorem for Compact Self-Adjoint Operators
Let $H$ be a Hilbert space. Let $T \in B_0 \left({H}\right)$ be a compact, self-adjoint operator. Then there exists a (possibly finite) sequence $\left({\lambda_n}\right)$ of distinct nonzero eigenvalues of $T$ such that: * $P_n P_m = P_m P_n = 0$ if $n \ne m$ * $\displaystyle \lim_{k \to \infty} \left\Vert{T - \sum_{n=1}^k \lambda_n P_n}\right\Vert = 0$, that is, $T = \displaystyle \sum_{n=1}^\infty \lambda_n P_n$ where $P_n$ is the orthogonal projection onto the eigenspace of $\lambda_n$, and $\left\Vert{\cdot}\right\Vert$ denotes the norm on bounded linear operators. {{refactor|Split corollaries to subpages}} === Corollary 1 === There exists a (possibly finite) sequence $\left({\mu_n}\right)$ of real numbers and a basis $E = \left({e_n}\right)$ for $\left({\operatorname{ker} T}\right)^\perp$ such that: :$\forall h \in H: Th = \displaystyle \sum_{n=1}^\infty \left\langle{h, e_n}\right\rangle_H \mu_n e_n$
Eigenspace for Normal Operator is Reducing Subspace
Let $H$ be a Hilbert space over $\Bbb F \in \left\{{\R, \C}\right\}$. Let $A \in B \left({H}\right)$ be a normal operator. Let $\lambda \in \Bbb F$. Then $\ker \left({A - \lambda}\right)$ is a reducing subspace for $A$. Here $\ker$ denotes kernel.
Eigenvalues of Self-Adjoint Operator are Real
Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a self-adjoint operator. Then all eigenvalues of $A$ are real.
Operator Diagonalizable iff Basis of Eigenvectors
Let $H$ be a Hilbert space. Let $A:H \to H$ be a linear operator on $H$. Then $A$ is diagonalizable iff there exists a basis $E$ of $H$, consisting of eigenvectors for $A$.
Existence of Hartogs Number
Let $S$ be a set. Then $S$ has a Hartogs number.
Left and Right Inverse Mappings Implies Bijection
Let $f: S \to T$ be a mapping. Let $f$ be such that: : $\exists g_1: T \to S: g_1 \circ f = I_S$ : $\exists g_2: T \to S: f \circ g_2 = I_T$ where both $g_1$ and $g_2$ are mappings. Then $f$ is a bijection.
Left and Right Inverses of Mapping are Inverse Mapping
Let $f: S \to T$ be a mapping such that: :$(1): \quad \exists g_1: T \to S: g_1 \circ f = I_S$ :$(2): \quad \exists g_2: T \to S: f \circ g_2 = I_T$ Then: : $g_1 = g_2 = f^{-1}$ where $f^{-1}$ is the inverse of $f$.
Operator Commuting with Diagonalizable Operator
Let $H$ be a Hilbert space. Let $A = \displaystyle \sum_{i \mathop \in I} \alpha_i P_i$ be a diagonalizable operator on $H$. Let $B \in \map B H$ be a bounded linear operator. Then the following are equivalent: :$(1): \quad A B = B A$ :$(2): \quad$ For all $i \in I$, $\Rng {P_i}$ is a reducing subspace for $B$ where $\Rng {P_i}$ denotes range.
Composition of Mappings is Composition of Relations
Let $S_1, S_2, S_3$ be sets. Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings such that the domain of $f_2$ is the same set as the codomain of $f_1$. Let $f_2 \circ f_1$ be the composite of $f_1$ and $f_2$. Let $f_1$ and $f_2$ be considered as relations on $S_1 \times S_2$ and $S_2 \times S_3$ respectively. Then $f_2 \circ f_1$ defined as the composition of relations coincides with the definition of $f_2 \circ f_1$ as the composition of mappings.
Null Sequence in Exponential Sequence
Let $\sequence {a_n}_{n \mathop \in \N} \in \C$ be a sequence of complex numbers such that: :$\displaystyle \lim_{n \mathop \to +\infty}a_n = 0$ Then: :$\displaystyle \lim_{n \mathop \to +\infty} \paren {1 + \dfrac {a_n} n}^n = 1$
Identity Mapping is Right Identity
Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Then: :$f \circ I_S = f$ where $I_S$ is the identity mapping on $S$, and $\circ$ signifies composition of mappings.
Diagonal Relation is Left Identity
Let $\RR \subseteq S \times T$ be a relation on $S \times T$. Then: :$\Delta_T \circ \RR = \RR$ where $\Delta_T$ is the diagonal relation on $T$, and $\circ$ signifies composition of relations.
Isomorphism Preserves Associativity
Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures. Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an isomorphism. Then $\circ$ is associative  $*$ is associative.
Isomorphism Preserves Commutativity
Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures. Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an isomorphism. Then $\circ$ is commutative  $*$ is commutative.
Isomorphism Preserves Inverses
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be algebraic structures. Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be an isomorphism. Let $\left({S, \circ}\right)$ have an identity $e_S$. Then $x^{-1}$ is an inverse of $x$ for $\circ$ iff $\phi \left({x^{-1}}\right)$ is an inverse of $\phi \left({x}\right)$ for $*$. That is, iff: : $\phi \left({x^{-1}}\right) = \left({\phi \left({x}\right)}\right)^{-1}$
Isomorphism Preserves Identity
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be algebraic structures. Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be an isomorphism. Then $\circ$ has an identity $e_S$  $\phi \left({e_S}\right)$ is the identity for $*$.
Non-Zero Natural Numbers under Multiplication form Commutative Semigroup
Let $\N_{>0}$ be the set of natural numbers without zero, that is, $\N_{>0} = \N \setminus \set 0$. The structure $\struct {\N_{>0}, \times}$ forms an infinite commutative semigroup.
Isomorphism Preserves Groups
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be algebraic structures. Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be an isomorphism. If $\left({S, \circ}\right)$ is a group, then so is $\left({T, *}\right)$.
Isomorphism Preserves Semigroups
Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures. Let $\phi: S \to T$ be an isomorphism. If $\struct {S, \circ}$ is a semigroup, then so is $\struct {T, *}$.
Left Cancellable iff Left Regular Representation Injective
Let $\struct {S, \circ}$ be an algebraic structure. Then $a \in S$ is left cancellable  the left regular representation $\map {\lambda_a} x$ is injective.
Right Cancellable iff Right Regular Representation Injective
Let $\struct {S, \circ}$ be an algebraic structure. Then $a \in S$ is right cancellable  the right regular representation $\map {\rho_a} x$ is injective.
Isomorphism Preserves Cancellability
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be algebraic structures. Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be an isomorphism. Then: : $a \in S$ is cancellable in $\left({S, \circ}\right)$ iff $\phi \left({a}\right) \in T$ is cancellable in $\left({T, *}\right)$.
Isomorphism Preserves Left Cancellability
Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures. Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an isomorphism. Then: :$a \in S$ is left cancellable in $\struct {S, \circ}$ : :$\map \phi a \in T$ is left cancellable in $\struct {T, *}$.
Isomorphism Preserves Right Cancellability
Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures. Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an isomorphism. Then: :$a \in S$ is right cancellable in $\struct {S, \circ}$  $\map \phi a \in T$ is right cancellable in $\struct {T, *}$.
Differentiation of Power Series
Let $\xi \in \R$ be a real number. Let $\left \langle {a_n} \right \rangle$ be a sequence in $\R$. Let $\displaystyle \sum_{m \mathop \ge 0} a_m \left({x - \xi}\right)^m$ be the power series in $x$ about the point $\xi$. Then within the interval of convergence: :$\displaystyle \frac {\mathrm d^n} {\mathrm d x^n} \sum_{m \mathop \ge 0} a_m \left({x - \xi}\right)^m = \sum_{m \mathop \ge n} a_m m^{\underline n} \left({x - \xi}\right)^{m - n}$ where $m^{\underline n}$ denotes the falling factorial.
General Associativity Theorem/Formulation 1
Let $\struct {S, \circ}$ be a semigroup. Let $\sequence {a_k}_{p + 1 \mathop \le k \mathop \le p + n}$ be a sequence of elements of $S$. Let $\sequence {r_k}_{0 \mathop \le k \mathop \le s}$ be a strictly increasing sequence of natural numbers such that $r_0 = p$ and $r_s = p+n$. Suppose: :$\displaystyle \forall k \in \closedint 1 s: b_k = \prod_{j \mathop = r_{k - 1} \mathop + 1}^{r_k} {a_j}$ Then: :$\displaystyle \forall n \in \N_{>0}: \prod_{k \mathop = 1}^s {b_k} = \prod_{k \mathop = p \mathop + 1}^{p \mathop + n} {a_k}$ That is: :$\displaystyle \forall n \in \N_{>0}: \prod_{k \mathop = 1}^s \paren {a_{r_{k - 1} + 1} \circ a_{r_{k - 1} + 2} \circ \ldots \circ a_{r_k} } = a_{p + 1} \circ \ldots \circ a_{p + n}$
General Associativity Theorem/Formulation 2
Let $n \in \N_{>0}$ and let $a_1, \ldots, a_n$ be elements of a set $S$. Let $\circ$ be an associative operation on $S$. Let the set $\map {P_n} {a_1, a_2, \ldots, a_n}$ be defined inductively by: :$\map {P_1} {a_1} = \set {a_1}$ :$\map {P_2} {a_1, a_2} = \set {a_1 \circ a_2}$ :$\map {P_n} {a_1, a_2, \ldots, a_n} = \set {x \circ y: x \in \map {P_r} {a_1, a_2, \ldots, a_r} \land y \in \map {P_s} {a_{r + 1}, a_{r + 2}, \ldots, a_{r + s} }, n = r + s}$ Then $\map {P_n} {a_1, a_2, \ldots, a_n}$ consists of a unique entity which we can denote $a_1 \circ a_2 \circ \ldots \circ a_n$.
Monoid of Self-Inverse Elements is Abelian Group
Let $\left({S, \circ}\right)$ be a monoid such that: :$\forall x \in S: x \circ x = e$ where $e$ is the identity element of $\left({S, \circ}\right)$. Then $\left({S, \circ}\right)$ is an abelian group.
Product is Right Identity Therefore Right Cancellable
Let $\left({S, \circ}\right)$ be a semigroup. Let $e_R \in S$  be a right identity of $S$. Let $a \in S$ such that: :$\exists b \in S: a \circ b = e_R$ Then $a$ is right cancellable in $\left({S, \circ}\right)$.
Integer Multiples Greater than Positive Integer Closed under Multiplication
Let $n \Z$ be the set of integer multiples of $n$. Let $p \in \Z: p \ge 0$ be a positive integer. Let $S \subseteq n \Z$  be defined as: :$S := \left\{{x \in n \Z: x > p}\right\}$ that is, the set of integer multiples of $n$ greater than $p$. Then the algebraic structure $\struct {S, \times}$ is closed under multiplication.
Non-Zero Complex Numbers Closed under Multiplication
The set of non-zero complex numbers is closed under multiplication.
Non-Zero Modulo Numbers Closed under Multiplication then Modulo is Prime
Let $\left({\Z_m, +_m, \times_m}\right)$ be the ring of integers modulo $m$ for $m > 1$. Let $\Z'_m$ be the set of non-zero integers modulo $m$. Let $\left({\Z_m, \times_m}\right)$ be closed under modulo multiplication. Then $m$ is prime.
Magma is Submagma of Itself
Let $\left({S, \circ}\right)$ be a magma. Then $\left({S, \circ}\right)$ is a submagma of itself.
Empty Set is Submagma of Magma
Let $\left({S, \circ}\right)$ be a magma. Then: : $\left({\varnothing, \circ}\right)$ is a submagma of $\left({S, \circ}\right)$ where $\varnothing$ is the empty set.
Max Operation on Toset is Semigroup
Let $\left({S, \preceq}\right)$ be a totally ordered set. Let $\max \left({x, y}\right)$ denote the max operation on $x, y \in S$. Then $\left({S, \max}\right)$ is a semigroup.
Min Semigroup is Commutative
Let $\left({S, \preceq}\right)$ be a totally ordered set.  Then the semigroup $\left({S, \min}\right)$ is commutative.
Min Semigroup is Idempotent
Let $\struct {S, \preceq}$ be a totally ordered set. Then the semigroup $\struct {S, \min}$ is an idempotent semigroup.
Max Operation on Woset is Monoid
Let $\struct {S, \preceq}$ be a well-ordered set. Let $\max \set {x, y}$ denote the max operation on $x, y \in S$. Then $\struct {S, \max}$ is a monoid. Its identity element is the smallest element of $S$.
Max Operation on Natural Numbers forms Monoid
Let $\left({\N, \max}\right)$ denote the algebraic structure formed from the natural numbers $\N$ and the max operation. Then $\left({\N, \max}\right)$ is a monoid. Its identity element is the zero.
Complex Modulus Function is Continuous
Let $z_0 \in \C$ be a complex number. Then the complex modulus function is continuous at $z_0$.
Euclidean Space is Banach Space
Let $m$ be a positive integer. Then the Euclidean space $\R^m$, along with the Euclidean norm, forms a Banach space over $\R$.