hcju/proofwikips huggingface dataset:
Linear Second Order ODE/y'' - y = 3 exp -x
The second order ODE: :$(1): \quad y'' - y = 3 e^{-x}$ has the general solution: :$y = C_1 e^x + C_2 e^{-x} - \dfrac {3 x e^{-x} } 2$
Elements of Geometric Sequence from One which Divide Later Elements
Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a geometric Sequence of integers. Let $a_0 = 1$. Let $m \in \Z_{> 0}$. Then: :$\forall r \in \set {0, 1, \ldots, m}: a_k \divides a_m$ where $\divides$ denotes divisibility. {{:Euclid:Proposition/IX/11}}
Elements of Geometric Sequence from One Divisible by Prime
Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a geometric sequence of integers. Let $a_0 = 1$. Let $p$ be a prime number such that: :$p \divides a_n$ where $\divides$ denotes divisibility. Then $p \divides a_1$. {{:Euclid:Proposition/IX/12}}
Divisibility of Elements of Geometric Sequence from One where First Element is Prime
Let $Q_n = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of length $n$ consisting of integers only. Let $a_0 = 1$. Let $a_1$ be a prime number. Then the only divisors of $a_n$ are $a_j$ for $j \in \set {1, 2, \ldots, n}$. {{:Euclid:Proposition/IX/13}}
Linear Second Order ODE/y'' - 2 y' + y = exp x
The second order ODE: :$(1): \quad y'' - 2 y' + y = e^x$ has the general solution: :$y = C_1 e^x + C_2 x e^x + \dfrac {x^2 e^x} 2$
Linear First Order ODE/y' - y = x^2
The linear first order ODE: :$(1): \quad \dfrac {\d y} {\d x} - y = x^2$ has the general solution: :$y = C e^x - \paren {x^2 + 2 x + 2}$
Linear Second Order ODE/y'' + y = exp -x cos x
The second order ODE: :$(1): \quad y'' + y = e^{-x} \cos x$ has the general solution: :$y = \dfrac {e^{-x} } 5 \paren {\cos x - 2 \sin x} + C_1 \sin x + C_2 \cos x$
Restriction of Homeomorphism is Homeomorphism
Let $T_1 = \left({S_1, \tau_1}\right)$, $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces. Let $f: S_1 \to S_2$ be a homeomorphism between $T_1$ and $T_2$. Let $S$ be a subset of $S_1$. Let $f {\restriction_{S \times f\left[{S}\right]}} : S \to f \left[{S}\right]$ be the restriction of $f$ to $S \times f \left[{S}\right]$. Let $S$ and $f \left[{S}\right]$ bear their respective subspace topologies. Then $f {\restriction_{S \times f \left[{S}\right]}}$ is a homeomorphism.
Disjoint Independent Events means One is Void
Let $A$ and $B$ be events in a probability space. Suppose $A$ and $B$ are: :disjoint :independent. Then either $\map \Pr A = 0$ or $\map \Pr B = 0$. That is, if two events are disjoint and independent, at least one of them can't happen.
Product of Rational Cuts is Rational Cut
Let $p \in\ Q$ and $q \in \Q$ be rational numbers. Let $p^*$ and $q^*$ denote the rational cuts associated with $p$ and $q$. Then: :$p^* q^* = \paren {p q}^*$ Thus the operation of multiplication on the set of rational cuts is closed.
Set of Rational Cuts forms Ordered Field
Let $\RR$ denote the set of rational cuts. Let $\struct {\RR, +, \times, \le}$ denote the ordered structure formed from $\RR$ and: :the operation $+$ of addition of cuts :the operation $\times$ of multiplication of cuts :the ordering $\le$ of cuts. Then $\struct {\RR, + \times, \le}$ is an ordered field.
Infimum of Subset of Real Numbers May or May Not be in Subset
Let $S \subset \R$ be a proper subset of the set $\R$ of real numbers. Let $S$ admit an infimum $m$. Then $m$ may or may not be an element of $S$.
Apotome is Irrational
Every apotome is irrational, i.e.: : $\displaystyle \forall a, b \in \set {x \in \R_{>0} : x^2 \in \Q}: \paren {\frac a b \notin \Q \land \paren {\frac a b}^2 \in \Q} \implies \paren {\paren {a - b} \notin \Q \land \paren {a - b}^2 \notin \Q}$ {{:Euclid:Proposition/X/73}}
Basis Condition for Coarser Topology/Corollary 1
If $\BB_1$ and $\BB_2$ satisfy: :$\forall U \in \BB_1: \forall x \in U: \exists V \in \BB_2: x \in V \subseteq U$ then $\tau_1$ is coarser than $\tau_2$.
Derivatives of PGF of Discrete Uniform Distribution
Let $X$ be a discrete random variable with the discrete uniform distribution with parameter $n$. Then the derivatives of the PGF of $X$ {{WRT|Differentiation}} $s$ are: :$\dfrac {\d^m} {\d s^m} \map {\Pi_X} s = \begin{cases} \displaystyle \dfrac 1 n \sum_{k \mathop = m}^n k^{\underline m} s^{k - m} & : m \le n \\ 0 & : k > n \end{cases}$ where $k^{\underline m}$ is the falling factorial.
Set of Doubletons of Natural Numbers is Countable
Let $S$ be the set defined as: :$S = \set {\set {n_1, n_2}: n_1, n_2 \in \N, n_1 \ne n_2}$ where $\N$ denotes the set of natural numbers. Then $S$ is countably infinite.
Frege Set Theory is Logically Inconsistent
The system of axiomatic set theory that is Frege set theory is inconsistent.
Empty Class is Unique
There is exactly one empty class.
Existence of Set is Equivalent to Existence of Empty Set
Let $V$ be a basic universe. Let $P$ be the axiom: :$V$ has at least one element. Then $P$ is equivalent to the axiom of the empty set: :The empty class $\O$ is a set.
Associativity on Four Elements
Let $\struct {S, \circ}$ be a semigroup. Let $a, b, c, d \in S$. Then: :$a \circ b \circ c \circ d$ gives a unique answer no matter how the elements are associated.
Intersection of Class is Subset of Intersection of Subclass
Let $V$ be a basic universe. Let $A$ and $B$ be classes of $V$: :$A \subseteq V, B \subseteq V$ such that it is not the case that $A = B = \O$. Let $\displaystyle \bigcap A$ and $\displaystyle \bigcap B$ denote the intersection of $A$ and intersection of $B$ respectively. Let $A$ be a subclass of $B$: :$A \subseteq B$ Then $\displaystyle \bigcap B$ is a subset of $\displaystyle \bigcap A$: :$\displaystyle \bigcap B \subseteq \displaystyle \bigcap A$
Frobenius's Theorem
An algebraic associative real division algebra $A$ is isomorphic to $\R, \C$ or $\Bbb H$.
Areas of Circles are as Squares on Diameters/Lemma
{{:Euclid:Proposition/XII/2/Lemma}}
Dominance Relation is Ordering
Let $S$ and $T$ be cardinals. Let $S \preccurlyeq T$ denote that $S$ is dominated by $T$. Let $\mathbb S$ be any set of cardinals. Then the relational structure $\struct {\mathbb S, \preccurlyeq}$ is an ordered set. That is, $\preccurlyeq$ is an ordering (at least partial) on $\mathbb S$.
Basic Universe is Inductive
Let $V$ be a basic universe. Then $V$ is an inductive class.
Image of Subset is Image of Restriction
Let $f: S \to T$ be a mapping. Let $X \subseteq S$. Let $f {\restriction_X}$ be the restriction of $f$ to $X$. Then: :$f \sqbrk X = \Img {f {\restriction_X} }$ where $\Img f$ denotes the image of $f$, defined as: :$\Img f = \set {t \in T: \exists s \in S: t = \map f s}$
Binomial Distribution PMF
The probability mass function (pmf) of a binomially distributed random variable $X$ is equal to: :$\displaystyle \Pr \left({X = x}\right) = \binom n x p^x(1-p)^{n-x}$ where $n$ is the number of trials and $p$ is the probability of success.
Successor Mapping of Peano Structure has no Fixed Point
Let $\PP = \struct {P, s, 0}$ be a Peano structure. Then: :$\forall n \in P: \map s n \ne n$ That is, the successor mapping has no fixed points.
Closed Set is F-Sigma Set
Let $T = \struct {S, \tau}$ be a topological space. Let $V$ be a closed set of $T$. Then $V$ is an $F_\sigma$ set of $T$.
Union of Open Intervals of Positive Reals is Set of Strictly Positive Reals
Let $\R_{> 0}$ be the set of strictly positive real numbers. For all $x \in \R_{> 0}$, let $A_x$ be the open real interval $\openint 0 x$. Then: :$\displaystyle \bigcup_{x \mathop \in \R_{> 0} } A_x = \R_{> 0}$
Intersection of Closed Intervals of Positive Reals is Zero
Let $\R_{> 0}$ be the set of strictly positive real numbers. For all $x \in \R_{> 0}$, let $B_x$ be the closed real interval $\closedint 0 x$. Then: :$\displaystyle \bigcap_{x \mathop \in \R_{> 0} } B_x = \set 0$
Minimally Closed Class under Progressing Mapping is Well-Ordered
$N$ is well-ordered under the inclusion relation.
Sum of Independent Poisson Random Variables is Poisson
Let $X$ and $Y$ be independent discrete random variables with: :$X \sim \Poisson {\lambda_1}$ and :$Y \sim \Poisson {\lambda_2}$ for some $\lambda_1, \lambda_2 \in \R_{> 0}$. Then their sum $X + Y$ is distributed: :$X + Y \sim \Poisson {\lambda_1 + \lambda_2}$
Order of General Linear Group over Galois Field
Let $\GF$ be a Galois field with $p$ elements. Then the order of the general linear group $\GL {n, \GF}$ is: :$\displaystyle \prod_{j \mathop = 1}^n \paren {p^n - p^{j - 1} }$
Existence of Latin Squares
For each $n \in \Z_{>0}$ there exists at least one Latin square of order $n$.
Lipschitz Condition implies Uniform Continuity
Let $\left({M_1, d_1}\right)$ and $\left({M_2, d_2}\right)$ be metric spaces. Let $g: M_1 \to M_2$ satisfy the Lipschitz condition. Then $g$ is uniformly continuous on $M_1$.
Repeated Composition of Injection is Injection
Let $S$ be a set. Let $f: S \to S$ be an injection. Let the sequence of mappings: :$f^0, f^1, f^2, \ldots, f^n, \ldots$ be defined as: :$\forall n \in \N: \map {f^n} x = \begin {cases} x & : n = 0 \\ \map f x & : n = 1 \\ \map f {\map {f^{n - 1} } x} & : n > 1 \end{cases}$ Then for all $n \in \N$, $f^n$ is an injection.
Renaming Mapping from Set of Mappings on Single Element
Let $X$ and $Y$ be sets. Let $E$ be the set of all mappings from $X$ to $Y$. Let $b \in X$. Let $\mathcal R \subseteq E \times E$ be the relation on $E$ defined as: :$\mathcal R := \set {\tuple {f, g} \in \mathcal R: \map f b = \map g b}$ Let $e_b: E / \mathcal R \to Y$ be the renaming mapping induced by $\mathcal R$. Then $e_b$ is a bijection.
Power of Random Variable with Continuous Uniform Distribution has Beta Distribution
Let $X \sim \ContinuousUniform 0 1$ where $\ContinuousUniform 0 1$ is the continuous uniform distribution on $\closedint 0 1$. Let $n$ be a positive real number.  Then: :$X^n \sim \BetaDist {\dfrac 1 n} 1$ where $\operatorname {Beta}$ is the beta distribution.
Variance of Linear Transformation of Random Variable
Let $X$ be a random variable.  Let $a, b$ be real numbers.  Then we have:  :$\var {a X + b} = a^2 \var X$ where $\var X$ denotes the variance of $X$.
Versed Sine Function is Even
The versed sine is an even function: :$\forall \theta \in \R: \map \vers {-\theta} = \vers \theta$
Supremum Metric on Continuous Real Functions is Metric
Let $\left[{a \,.\,.\, b}\right] \subseteq \R$ be a closed real interval. Let $\mathscr C \left[{a \,.\,.\, b}\right]$ be the set of all continuous functions $f: \left[{a \,.\,.\, b}\right] \to \R$. Let $d$ be the supremum metric on  $\mathscr C \left[{a \,.\,.\, b}\right]$. Then $d$ is a metric.
P-Norm of Real Sequence is Strictly Decreasing Function of P
Let $p \ge 1$ be a real number. Let $\ell^p$ denote the $p$-sequence space. Let $\mathbf x = \sequence {x_n} \in \ell^p$. Suppose $\mathbf x$ is not a sequence of zero elements. Let $\norm {\mathbf x}_p$ denote the $p$-norm. Then the mapping $p \to \norm {\mathbf x}_p$ is strictly decreasing  $p$.
Extension of Half-Range Fourier Cosine Function to Symmetric Range
Let $\map f x$ be a real function defined on the interval $\openint 0 \lambda$. Let $\map f x$ be represented by the half-range Fourier cosine series $\map S x$: :$\map f x \sim \map S x = \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty a_n \cos \frac {n \pi x} \lambda$ where for all $n \in \Z_{> 0}$: :$a_n = \displaystyle \frac 2 \lambda \int_0^\lambda \map f x \cos \frac {n \pi x} \lambda \rd x$ Then $\map S x$ also represents the extension to the even function $g: \openint {-\lambda} \lambda \to \R$ of $f$, defined as: :$\forall x \in \openint {-\lambda} \lambda: \map g x = \begin {cases} \map f x & : x > 0 \\ \map f {-x} & : x < 0 \\ \displaystyle \lim_{x \mathop \to 0} \map g x & : x = 0 \end {cases}$
P-adic Metric is Metric
Let $p \in \N$ be a prime. Let $\norm{\,\cdot\,}_p: \Q \to \R_{\ge 0}$ be the $p$-adic norm on $\Q$. Let $d_p$ be the $p$-adic metric on $\Q$: :$\forall x, y \in \Q: \map {d_p} {x, y} = \norm{x - y}_p$ Then $d_p$ is a metric.
Sum over Integers of Sine of n + alpha of theta over n + alpha
For $0 < \theta < 2 \pi$: :$\displaystyle \sum_{n \mathop \in \Z} \dfrac {\sin \paren {n + \alpha} \theta} {n + \alpha} = \pi$
Mittag-Leffler Expansion for Cotangent Function/Real Domain
:$\displaystyle \dfrac 1 \alpha + \sum_{n \mathop \ge 1} \dfrac {2 \alpha} {\alpha^2 - n^2} = \pi \cot \pi \alpha$
Sum of Cosines of Arithmetic Sequence of Angles/Formulation 1
{{begin-eqn}} {{eqn | l = \sum_{k \mathop = 0}^n \map \cos {\theta + k \alpha}       | r = \cos \theta + \map \cos {\theta + \alpha} + \map \cos {\theta + 2 \alpha} + \map \cos {\theta + 3 \alpha} + \dotsb }} {{eqn | r = \frac {\map \sin {\alpha \paren {n + 1} / 2} } {\map \sin {\alpha / 2} } \map \cos {\theta + \frac {n \alpha} 2} }} {{end-eqn}}
Count of Binary Operations Without Identity
Let $S$ be a set whose cardinality is $n$. The number $N$ of possible different binary operations which do not have an identity element that can be applied to $S$ is given by: :$N = n^{\left({\left({n-1}\right)^2 + 1}\right)} \left({n^{2 \left({n-1}\right)} - 1}\right)$
Addition of Coordinates on Cartesian Plane under Chebyshev Distance is Continuous Function
Let $\R^2$ be the real number plane. Let $d_\infty$ be the Chebyshev distance on $\R^2$. Let $f: \R^2 \to \R$ be the real-valued function defined as: :$\forall \left({x_1, x_2}\right) \in \R^2: f \left({x_1, x_2}\right) = x_1 + x_2$ Then $f$ is continuous.
Equivalence of Definitions of Order of Group Element
{{TFAE|def = Order of Group Element}} Let $G$ be a group whose identity is $e$. Let $x \in G$.
Maximum Rule for Continuous Functions
Let $\struct {S, \tau}$ be a topological space. Let $f, g: S \to \R$ be continuous real-valued functions. Let $\max \set {f, g}: S \to \R$ denote the pointwise maximum of $f$ and $g$. Then: :$\max \set {f, g}$ is continuous.
Continuity of Mapping to Cartesian Product under Chebyshev Distance
Let $M_1 = \left({A_1, d_1}\right), M_2 = \left({A_2, d_2}\right), \ldots, M_n = \left({A_n, d_n}\right)$ be metric spaces. Let $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \ldots, A_n$. Let $d_\infty: \mathcal A \times \mathcal A \to \R$ be the Chebyshev distance on $\mathcal A$: : $\displaystyle d_\infty \left({x, y}\right) = \max_{i \mathop = 1}^n \left\{ {d_i \left({x_i, y_i}\right)}\right\}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$. For all $i \in \left\{ {1, 2, \ldots, n}\right\}$, let $\operatorname{pr}_i: \mathcal A \to A_i$ be the $i$th projection on $\mathcal A$: :$\forall a \in \mathcal A: \operatorname{pr}_i \left({a}\right) = a_i$ where $a = \left({a_1, a_2, \ldots, a_n}\right) \in \mathcal A$. Let $M' = \left({X, d'}\right)$ be a metric space. Let $f: X \to \mathcal A$ be a mapping. Then $f$ is continuous on $X$  each of $\operatorname{pr}_i \circ f: X \to A_i$ is continuous on $X$.
Union of Mappings with Disjoint Domains is Mapping
Let $S_1, S_2, T_1, T_2$ be sets. Let $f: S_1 \to T_1$ and $g: S_2 \to T_2$ be mappings. Let $h = f \cup g$ be their union. If $S_1 \cap S_2 = \O$, then $h: S_1 \cup S_2 \to T_1 \cup T_2$ is a mapping whose domain is $S_1 \cup S_2$.
Natural Number m is Less than n iff m is an Element of n
Let $\omega$ be the set of natural numbers defined as the von Neumann construction. Let $m, n \in \omega$. Then: :$m < n \iff m \in n$ That is, every natural number is the set of all smaller natural numbers.
Limit of Image of Sequence/Real Number Line
Let $f$ be a real function which is continuous on the interval $\Bbb I$. Let $\left \langle {x_n} \right \rangle$ be a sequence of points in $\Bbb I$ such that: : $\displaystyle \lim_{n \mathop \to \infty} x_n = \xi$ where: : $(1): \quad \xi \in \Bbb I$ : $(2): \quad \displaystyle \lim_{n \mathop \to \infty} x_n$ denotes the limit of $x_n$. Then: : $\displaystyle \lim_{n \mathop \to \infty} f \left({x_n}\right) = f \left({\xi}\right)$ That is: : $\displaystyle \lim_{n \mathop \to \infty} f \left({x_n}\right) = f \left({\lim_{n \mathop \to \infty} x_n}\right)$
Like Electric Charges Repel
Let $a$ and $b$ be stationary particles, each carrying an electric charge of $q_a$ and $q_b$ respectively. Let $q_a$ and $q_b$ be of the same sign. That is, let $q_a$ and $q_b$ be like charges. Then the forces exerted by $a$ on $b$, and by $b$ on $a$, are such as to cause $a$ and $b$ to repel each other.
P-Sequence Space with P-Norm forms Banach Space
Let $\ell^p$ be a p-sequence space. Let $\norm {\, \cdot \,}_p$ be a p-norm. Then $\struct {\ell^p, \norm {\, \cdot \,}_p}$ is a Banach space.
Neighbourhood of Point Contains Point of Subset iff Distance is Zero
Let $M = \struct {X, d}$ be a metric space. Let $A \subseteq X$ be a non-empty subset of $X$. Let $x \in X$. Then every neighborhood of $x$ contains a point of $A$ : :$\map d {x, A} = 0$ where $\map d {x, A}$ denotes the distance from $x$ to $A$.
Set is Open iff Union of Open Balls
Let $M = \struct {A, d}$ be a metric space. Let $U \subseteq A$. Then $U$ is open in $M$  it is a union of open balls.
Empty Set is Open and Closed in Metric Space
Let $M = \left({A, d}\right)$ be a metric space. Then the empty set $\varnothing$ is both open and closed in $M$.
Equivalence of Definitions of Independent Subgroups
{{TFAE|def = Independent Subgroups}} Let $G$ be a group whose identity is $e$. Let $\left \langle {H_n} \right \rangle$ be a sequence of independent subgroups of $G$.
Condition for Factoring of Quotient Mapping between Modulo Addition Groups
Let $m, n \in \Z_{>0}$ be strictly positive integers. Let $\struct {\Z, +}$ denote the additive group of integers. Let $\struct {\Z_m, +_m}$ and $\struct {\Z_n, +_n}$ denote the additive groups of integers modulo $m$ and $n$ respectively. Let $f: \Z \to \Z_n$ be the quotient epimorphism from $\struct {\Z, +}$ to $\struct {\Z_n, +_n}$. Let $q: \Z \to \Z_m$ be the quotient epimorphism from $\struct {\Z, +}$ to $\struct {\Z_m, +_m}$. Then Then $N \subseteq K$ Then: :there exists a group homomorphism $\psi: \struct {\Z_m, +_m} \to \struct {\Z_n, +_n}$   :$m \divides n$ where $\divides$ denotes divisibility.
Construction of Inverse Completion/Quotient Structure
Let the quotient structure defined by $\boxtimes$ be: : $\displaystyle \left({T', \oplus'}\right) := \left({\frac {S \times C} \boxtimes, \oplus_\boxtimes}\right)$ where $\oplus_\boxtimes$ is the operation induced on $\displaystyle \frac {S \times C} \boxtimes$ by $\oplus$. === Quotient Structure is Commutative Semigroup === {{:Construction of Inverse Completion/Quotient Structure is Commutative Semigroup}} === Quotient Mapping is Injective === {{:Construction of Inverse Completion/Quotient Mapping is Injective}} === Quotient Mapping is Monomorphism === {{:Construction of Inverse Completion/Quotient Mapping is Monomorphism}} === Image of Quotient Mapping is Subsemigroup === {{:Construction of Inverse Completion/Image of Quotient Mapping is Subsemigroup}} === Quotient Mapping to Image is Isomorphism === {{:Construction of Inverse Completion/Quotient Mapping to Image is Isomorphism}} === Image of Cancellable Elements in Quotient Mapping === {{:Construction of Inverse Completion/Quotient Mapping/Image of Cancellable Elements}}
Inclusion Mapping on Metric Space is Continuous
Let $M = \left({A, d}\right)$ be a metric space. Let $\left({H, d_H}\right)$ be a metric subspace of $M$. Then the inclusion mapping $i_H: H \to A$ is continuous.
Set of Transpositions is not Subgroup of Symmetric Group
Let $S$ be a finite set with $n$ elements such that $n > 2$. Let $G = \struct {\map \Gamma S, \circ}$ denote the symmetric group on $S$. Let $H \subseteq G$ denote the set of all transpositions of $S$ along with the identity mapping which moves no elements of $S$. Then $H$ does not form a subgroup of $G$.
Homomorphism from Reals to Circle Group/Corollary
Let $\struct {\R, +}$ be the additive group of real numbers. Let $\struct {C_{\ne 0}, \times}$ be the multiplicative group of complex numbers. Let $\phi: \struct {\R, +} \to \struct {C_{\ne 0}, \times}$ be the mapping defined as: :$\forall x \in \R: \map \phi x = \cos x + i \sin x$ Then $\phi$ is a (group) homomorphism.
Projection onto Ideal of External Direct Sum of Rings
Let $\left({R_1, +_1, \circ_1}\right), \left({R_2, +_2, \circ_2}\right), \ldots, \left({R_n, +_n, \circ_n}\right)$ be rings. Let $\displaystyle \left({R, +, \circ}\right) = \prod_{k \mathop = 1}^n \left({R_k, +_k, \circ_k}\right)$ be their direct product. For each $k \in \left[{1 \,.\,.\, n}\right]$, let: :$R'_k = \left\{{\left({x_1, \ldots, x_n}\right) \in R: \forall j \ne k: x_j = 0}\right\}$ Let $\operatorname{pr}_k: R \to R'_k$ be the projection on the $k$th coordinate of $\left({R, +, \circ}\right)$ onto $R'_k$. Then $\operatorname{pr}_k$ is an epimorphism.
Edgeless Graph is Bipartite
Let $N_n$ denote the edgeless graph with $n$ vertices. Then $N_n$ is a bipartite graph.
Set of Subfields forms Complete Lattice
Let $\struct {F, +, \circ}$ be a field. Let $\mathbb F$ be the set of all subfields of $F$. Then $\struct {\mathbb F, \subseteq}$ is a complete lattice.
Simple Graph of Maximum Size is Complete Graph
Let $G$ be a simple graph of order $n$ such that $n \ge 1$. Let $G$ have the largest size of all simple graphs of order $n$. Then: :$G$ is the complete graph $K_n$ :its size is $\dfrac {n \paren {n - 1} } 2$.
Sum of Integrals on Complementary Sets
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $A, E \in \Sigma$ with $A \subseteq E$. Let $f$ be a $\mu$-integrable function on $X$. Then  :$\displaystyle \int_E f \rd \mu = \int_A f \rd \mu + \int_{E \mathop \setminus A} f \rd \mu$
Characteristics of Cycle Graph
Let $G = \struct {V, E}$ be an (undirected) graph whose order is greater than $2$. Then $G$ is a cycle graph : :$G$ is connected :every vertex of $G$ is adjacent to $2$ other vertices :every edge of $G$ is adjacent to $2$ other edges.
Neighborhood of Point in Metrizable Space contains Closed Neighborhood
Let $T = \struct {S, \tau}$ be a metrizable topological space. Let $x \in S$ be an arbitrary point of $T$. Let $N$ be a neighborhood of $x$. Then $N$ has as a subset a neighborhood $V$ of $x$ such that $V$ is closed.
Induced Neighborhood Space is Neighborhood Space
Let $S$ be a set. Let $\tau$ be a topology on $S$, thus forming the topological space $\left({S, \tau}\right)$. Let $\left({S, \mathcal N}\right)$ be the neighborhood space induced by $\left({S, \tau}\right)$. Then $\left({S, \mathcal N}\right)$ is a neighborhood space.
Order 1 Simple Graph is Unique up to Isomorphism
Let $G_1 = \struct {\map V {G_1}, \map E {G_1} }$ and $G_2 = \struct {\map V {G_2}, \map E {G_2} }$ be simple graphs of order $1$. Then $G_1$ and $G_2$ are isomorphic.
Definition of Polynomial from Polynomial Ring over Sequence
Let $\struct {R, +, \circ}$ be a ring with unity. Let $\struct {P \sqbrk R, \oplus, \odot}$ be the polynomial ring over the set of all sequences in $R$: :$P \sqbrk R = \set {\sequence {r_0, r_1, r_2, \ldots} }$ where the operations $\oplus$ and $\odot$ on $P \sqbrk R$ be defined as: {{:Definition:Operations on Polynomial Ring of Sequences}} Let $\struct {R \sqbrk X, +, \circ}$ be the ring of polynomials over $R$ in $X$. {{explain|Strictly speaking the definition in terms of Definition:Polynomial Form is needed here, with $X$ specifically being an Definition:Indeterminate (Polynomial Theory) or Definition:Transcendental over Integral Domain.}} Then $\struct {R \sqbrk X, +, \circ}$ and $\struct {P \sqbrk R, \oplus, \odot}$ are isomorphic.
Bertrand's Theorem
Let $U: \R_{>0} \to \R$ be analytic for $r > 0$. Let $M > 0$ be a nonvanishing angular momentum such that a stable circular orbit exists. Suppose that every orbit sufficiently close to the circular orbit is closed. Then $U$ is either $k r^2$ or $-\dfrac k r$ (for $k > 0$) up to an additive constant.
Cube Root of 2 is Irrational
:$\sqrt [3] 2$ is irrational.
Integers under Multiplication form Countably Infinite Semigroup
The set of integers under multiplication $\struct {\Z, \times}$ is a countably infinite semigroup.
Third Derivative of Natural Logarithm Function
Let $\ln x$ be the natural logarithm function. Then: :$\map {D^3_x} {\ln x} = \dfrac 2 {x^3}$