hcju/proofwikips huggingface dataset:
Definite Integral to Infinity of Cosine m x over x Squared plus a Squared
:$\displaystyle \int_0^\infty \frac {\cos m x} {x^2 + a^2} \rd x = \frac \pi {2 a} e^{-m a}$
Combination Theorem for Continuous Mappings/Normed Division Ring
Let $\struct{S, \tau_{_S}}$ be a topological space. Let $\struct{R, +, *, \norm{\,\cdot\,}}$ be a normed division ring. Let $\tau_{_R}$ be the topology induced by the norm $\norm{\,\cdot\,}$. Let $\lambda \in R$. Let $f,g : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ be continuous mappings. Let $U = S \setminus \set{x : \map g x = 0}$ Let $g^{-1} : U \to R$ denote the mapping defined by: :$\forall x \in U : \map {g^{-1}} x = \map g x^{-1}$ Let $\tau_{_U}$ be the subspace topology on $U$. Then the following results hold: === Sum Rule === {{:Combination Theorem for Continuous Mappings/Normed Division Ring/Sum Rule}} === Translation Rule === {{:Combination Theorem for Continuous Mappings/Normed Division Ring/Translation Rule}} === Negation Rule === {{:Combination Theorem for Continuous Mappings/Normed Division Ring/Negation Rule}} === Product Rule === {{:Combination Theorem for Continuous Mappings/Normed Division Ring/Product Rule}} === Multiple Rule === {{:Combination Theorem for Continuous Mappings/Normed Division Ring/Multiple Rule}} === Inverse Rule === {{:Combination Theorem for Continuous Mappings/Normed Division Ring/Inverse Rule}}
Combination Theorem for Continuous Mappings/Normed Division Ring/Sum Rule
:$f + g: \struct {S, \tau_{_S} } \to \struct{R, \tau_{_R} }$ is continuous.
Combination Theorem for Continuous Mappings/Normed Division Ring/Multiple Rule
:$\lambda * f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous :$f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.
Combination Theorem for Continuous Mappings/Normed Division Ring/Product Rule
:$f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.
Combination Theorem for Continuous Mappings/Normed Division Ring/Inverse Rule
:$g^{-1}: \struct {U, \tau_{_U} } \to \struct {R, \tau_{_R} }$ is continuous.
Derivative of Cosine Integral Function
:$\displaystyle \frac \d {\d x} \paren {\map \Ci x} = -\frac {\cos x} x$
Derivative of Exponential Integral Function
:$\displaystyle \frac \d {\d x} \paren {\map \Ei x} = -\frac {e^{-x} } x$
Derivative of Sine Integral Function
:$\displaystyle \frac \d {\d x} \paren {\map \Si x} = \frac {\sin x} x$
Primitive of Exponential Integral Function
:$\displaystyle \int \map \Ei x \rd x = x \map \Ei x - e^{-x} + C$
Direct Image Mapping of Domain is Image Set of Mapping
Let $S$ and $T$ be sets. Let $\powerset S$ and $\powerset T$ be their power sets. Let $f \subseteq S \times T$ be a mapping from $S$ to $T$. Let $f^\to: \powerset S \to \powerset T$ be the direct image mapping of $f$: :$\forall X \in \powerset S: \map {f^\to} X = \begin {cases} \set {t \in T: \exists s \in X: \map f s = t} & : X \ne \O \\ \O & : X = \O \end {cases}$ Then: :$\map {f^\to} S = \Img f$ where $\Img f$ is the image set of $f$.
Direct Image Mapping of Domain is Image Set of Relation
Let $S$ and $T$ be sets. Let $\powerset S$ and $\powerset T$ be their power sets. Let $\mathcal R \subseteq S \times T$ be a relation on $S \times T$. Let $\mathcal R^\to: \powerset S \to \powerset T$ be the direct image mapping of $\mathcal R$: :$\forall X \in \powerset S: \map {\mathcal R^\to} X = \begin {cases} \set {t \in T: \exists s \in X: \tuple {x, t} \in \mathcal R} & : X \ne \O \\ \O & : X = \O \end {cases}$ Then: :$\map {\mathcal R^\to} {\Dom {\mathcal R} } = \Img {\mathcal R}$ where: :$\Dom {\mathcal R}$ is the domain of $\mathcal R$ :$\Img {\mathcal R}$ is the image set of $\mathcal R$.
Inverse Image Mapping of Codomain is Preimage Set of Mapping
Let $S$ and $T$ be sets. Let $\powerset S$ and $\powerset T$ be their power sets. Let $f \subseteq S \times T$ be a mapping from $S$ to $T$. Let $f^\gets: \powerset T \to \powerset S$ be the inverse image mapping of $f$: :$\forall Y \in \powerset T: \map {f^\gets} Y = \begin {cases} \set {s \in S: \exists t \in Y: \map f s = t} & : \Img f \cap Y \ne \O \\ \O & : \Img f \cap Y = \O \end {cases}$ Then: :$\map {f^\gets} T = \Preimg f$ where $\Preimg f$ is the preimage set of $f$.
Inverse Image Mapping of Codomain is Preimage Set of Relation
Let $S$ and $T$ be sets. Let $\powerset S$ and $\powerset T$ be their power sets. Let $\mathcal R \subseteq S \times T$ be a relation on $S \times T$. Let $\mathcal R^\gets: \powerset T \to \powerset S$ be the inverse image mapping of $\mathcal R$: :$\forall X \in \powerset S: \map {\mathcal R^\to} X = \begin {cases} \set {t \in T: \exists s \in X: \tuple {x, t} \in \mathcal R} & : X \ne \O \\ \O & : X = \O \end {cases}$ Then: :$\map {\mathcal R^\gets} T = \Preimg {\mathcal R}$ where $\Preimg {\mathcal R}$ is the preimage of $\mathcal R$.
Direct Image Mapping of Mapping is Mapping
Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping on $S \times T$. Let $f^\to: \powerset S \to \powerset T$ be the direct image mapping of $f$: :$\forall X \in \powerset S: \map {f^\to} X = \set {t \in T: \exists s \in X: \map f s = t}$ Then $f^\to$ is indeed a mapping.
Inverse Image Mapping of Mapping is Mapping
Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping from $S$ to $T$. Let $f^\gets$ be the inverse image mapping of $f$: :$f^\gets: \powerset T \to \powerset S: \map {f^\gets} Y = f^{-1} \sqbrk Y$ Then $f^\gets$ is indeed a mapping.
Direct Image Mapping of Mapping is Empty iff Argument is Empty
Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping from $S$ to $T$. Let $f^\to$ be the direct image mapping of $f$: :$f^\to: \powerset S \to \powerset T: \map {f^\to} X = \set {t \in T: \exists s \in X: \map f s = t}$ Then: :$\map {f^\to} X = \O \iff X = \O$
Derivative of Fresnel Sine Integral Function
:$\displaystyle \frac {\d \operatorname S} {\d x} = \sqrt {\frac 2 \pi} \sin x^2$
Derivative of Fresnel Cosine Integral Function
:$\displaystyle \frac {\d \operatorname C} {\d x} = \sqrt {\frac 2 \pi} \cos x^2$
Power Series Expansion for Fresnel Cosine Integral Function
:$\displaystyle \map {\operatorname C} x = \sqrt {\frac 2 \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{4 n + 1} } {\paren {4 n + 1} \paren {2 n}!}$
Fresnel Sine Integral Function is Odd
:$\map {\operatorname S} {-x} = -\map {\operatorname S} x$
Fresnel Sine Integral Function of Zero
:$\map {\operatorname S} 0 = 0$
Fresnel Cosine Integral Function is Odd
:$\map {\operatorname C} {-x} = -\map {\operatorname C} x$
Fresnel Cosine Integral Function of Zero
:$\map {\operatorname C} 0 = 0$
Direct Image Mapping of Left-Total Relation is Empty iff Argument is Empty
Let $S$ and $T$ be sets. Let $\mathcal R: S \to T$ be a left-total relation on $S \times T$. Let $\mathcal R^\to$ be the direct image mapping of $\mathcal R$: :$\mathcal R^\to: \powerset S \to \powerset T: \map {\mathcal R^\to} X = \set {t \in T: \exists s \in X: \tuple {s, t} \in \mathcal R}$ Then: :$\map {\mathcal R^\to} X = \O \iff X = \O$
Limit to Infinity of Fresnel Cosine Integral Function
:$\displaystyle \lim_{x \mathop \to \infty} \map {\operatorname C} x = \frac 1 2$
Composition of Direct Image Mappings of Mappings
Let $A, B, C$ be non-empty sets. Let $f: A \to B$ and $g: B \to C$ be mappings. Let: :$f^\to: \powerset A \to \powerset B$ and :$g^\to: \powerset B \to \powerset C$ be the direct image mappings of $f$ and $g$. Then: :$\paren {g \circ f}^\to = g^\to \circ f^\to$
Composition of Inverse Image Mappings of Mappings
Let $A, B, C$ be non-empty sets. Let $f: A \to B, g: B \to C$ be mappings. Let: :$f^\gets: \powerset B \to \powerset A$ and :$g^\gets: \powerset C \to \powerset B$ be the inverse image mappings of $f$ and $g$. Then: :$\paren {g \circ f}^\gets = f^\gets \circ g^\gets$
Unit of Ring of Mappings iff Image is Subset of Ring Units
Let $\struct {R, +, \circ}$ be a ring with unity $1_R$. Let $U_R$ be the set of units in $R$. Let $S$ be a set. Let $\struct {R^S, +', \circ'}$ be the ring of mappings on the set of mappings $R^S$. Then: :$f \in R^S$ is a unit of $R^S$  $\Img f \subseteq U_R$ where $\Img f$ is the image of $f$. In this case, the inverse of $f$ is the mapping $f^{-1} : S \to U_R$ defined by: :$\forall x \in S : \map {f^{-1} } x = \map f x^{-1}$
Relative Complement Mapping on Powerset is Bijection
Let $S$ be a set. Let $\complement_S: \powerset S \to \powerset S$ denote the relative complement mapping on the power set of $S$. Then $\complement_S$ is a bijection. Thus each $T \subseteq S$ is in one-to-one correspondence with its relative complement.
Power of One plus x in terms of Gaussian Hypergeometric Function
:$\displaystyle {}_2 \map {F_1} {-p, 1; 1; -x} = \paren {1 + x}^p$
Laplace Transform of Exponential times Sine
:$\map {\laptrans {e^{b t} \sin a t} } s = \dfrac a {\paren {s - b}^2 + a^2}$
Inverse Image of Direct Image of Inverse Image equals Inverse Image Mapping
Let $f: S \to T$ be a mapping. Let: :$f^\to: \powerset S \to \powerset T$ denote the direct image mapping of $f$ :$f^\gets: \powerset T \to \powerset S$ denote the inverse image mapping of $f$ where $\powerset S$ denotes the power set of $S$. Then: :$f^\gets \circ f^\to \circ f^\gets = f^\gets$ where $\circ$ denotes composition of mappings.
Graph of Real Function in Cartesian Plane intersects Vertical at One Point
Let $f: \R \to \R$ be a real function. Let its graph be embedded in the Cartesian plane $\mathcal C$: :520px Every vertical line through a point $a$ in the domain of $f$ intersects the graph of $f$ at exactly one point $P = \tuple {a, \map f a}$.
Equation of Vertical Line
Let $\mathcal L$ be a vertical line embedded in the Cartesian plane $\mathcal C$. Then the equation of $\mathcal L$ can be given by: :$x = a$ where $\tuple {a, 0}$ is the point at which $\mathcal L$ intersects the $x$-axis. :520px
Equation of Horizontal Line
Let $\mathcal L$ be a horizontal line embedded in the Cartesian plane $\mathcal C$. Then the equation of $\mathcal L$ can be given by: :$y = b$ where $\tuple {0, b}$ is the point at which $\mathcal L$ intersects the $y$-axis. :520px
Unit of Ring of Mappings iff Image is Subset of Ring Units/Image is Subset of Ring Units implies Unit of Ring of Mappings
Let $\struct {R, +, \circ}$ be a ring with unity $1_R$. Let $U_R$ be the set of units in $R$. Let $S$ be a set. Let $\struct {R^S, +', \circ'}$ be the ring of mappings on the set of mappings $R^S$. Let $\Img f \subseteq U_R$ where $\Img f$ is the image of $f$. Then: :$f \in R^S$ is a unit of $R^S$ and the inverse of $f$ is the mapping $f^{-1} : S \to U_R$ defined by: :$\forall x \in S : \map {f^{-1}} {x} = \map f x^{-1}$
Unit of Ring of Mappings iff Image is Subset of Ring Units/Unit of Ring of Mappings implies Image is Subset of Ring Units
Let $\struct {R, +, \circ}$ be a ring with unity $1_R$. Let $U_R$ be the set of units in $R$. Let $S$ be a set. Let $\struct {R^S, +', \circ'}$ be the ring of mappings on the set of mappings $R^S$. Let $f \in R^S$ be a unit of $R^S$. Then: :$\Img f \subseteq U_R$ where $\Img f$ is the image of $f$. In which case, the inverse of $f$ is the mapping $f^{-1} : S \to U_R$ defined by: :$\forall x \in S : \map {f^{-1} } x = \map f x^{-1}$
Graph of Real Surjection in Coordinate Plane intersects Every Horizontal Line
Let $f: \R \to \R$ be a real function which is surjective. Let its graph be embedded in the Cartesian plane $\mathcal C$: :520px Every horizontal line through a point $b$ in the codomain of $f$ intersects the graph of $f$ on at least one point $P = \tuple {a, b}$ where $b = \map f a$.
Graph of Real Injection in Coordinate Plane intersects Horizontal Line at most Once
Let $f: \R \to \R$ be a real function which is injective. Let its graph be embedded in the Cartesian plane $\mathcal C$: :520px Let $\mathcal L$ be a horizontal line through a point $b$ in the codomain of $f$. Then $\mathcal L$ intersects the graph of $f$ on at most one point $P = \tuple {a, b}$ where $b = \map f a$.
Graph of Real Bijection in Coordinate Plane intersects Horizontal Line at One Point
Let $f: \R \to \R$ be a real function which is bijective. Let its graph be embedded in the Cartesian plane $\mathcal C$: :520px Every horizontal line through a point $b$ in the codomain of $f$ intersects the graph of $f$ on exactly one point $P = \tuple {a, b}$ where $b = \map f a$.
Composition of 3 Mappings where Pairs of Mappings are Bijections
Let $A$, $B$, $C$ and $D$ be sets. Let: :$f: A \to B$ :$g: B \to C$ :$h: C \to D$ be mappings. Let $g \circ f$ and $h \circ g$ be bijections. Then $f$, $g$ and $h$ are all bijections.
Mapping Composed with Bijection forming Bijection is Bijection
Let $A$, $B$ and $C$ be sets. Let $f: A \to B$ and $g: B \to C$ be mappings. Let the composite mapping $g \circ f$ be a bijection. Let either $f$ or $g$ be a bijection. Then both $f$ and $g$ are bijections.
Composite of Three Mappings in Cycle forming Injections and Surjection
Let $A$, $B$ and $C$ be non-empty sets. Let $f: A \to B$, $g: B \to C$ and $h: C \to A$ be mappings. Let the following hold: :$h \circ g \circ f$ is an injection :$f \circ h \circ g$ is an injection :$g \circ f \circ h$ is a surjection. where: :$g \circ f$ (and so on) denote composition of mappings. Then each of $f$, $g$ and $h$ are bijections.
Component Mappings of Set Coproduct are Injective
Let $S_1$ and $S_2$ be sets. Let $\struct {C, i_1, i_2}$ be a coproduct of $S_1$ and $S_2$. Then $i_1$ and $i_2$ are injections.
Existence of Bijection between Coproducts of two Sets
Let $S_1$ and $S_2$ be sets. Let $\struct {C, i_1, i_2}$ and $\struct {D, j_1, j_2}$ be two coproducts on $S_1$ and $S_2$. Then there exists a unique bijection $\theta: D \to C$ such that: :$\theta \circ j_i = i_1$ :$\theta \circ j_2 = i_2$
Number of Friday 13ths in a Year
In any given year, there are between $1$ and $3$ (inclusive) months in which the $13$th falls on a Friday.
Equivalence Relation on Natural Numbers such that Quotient is Power of Two/Equivalence Class Contains 1 Odd Number
Let $\eqclass n \alpha$ be the $\alpha$-equivalence class of a natural number $n$. Then $\eqclass n \alpha$ contains exactly $1$ odd number.
Equivalence Relation on Natural Numbers such that Quotient is Power of Two/One of Pair of Equivalent Elements is Divisor of the Other
Let $c, d \in \N$ such that $c \mathrel \alpha d$. Then either: :$c \divides d$ or: :$d \divides c$ where $\divides$ denotes divisibility.
Exists Divisor in Set of n+1 Natural Numbers no greater than 2n
Let $S$ be a set of $n + 1$ non-non-zero natural numbers all less than or equal to $2 n$. Then there exists $a, b \in S$ such that :$a \divides b$ where $\divides$ denotes divisibility.
Equality of Squares Modulo Integer is Equivalence Relation
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\RR_n$ be the relation on the set of integers $\Z$ defined as: :$\forall x, y \in \Z: x \mathrel {\RR_n} y \iff x^2 \equiv y^2 \pmod n$ Then $\RR_n$ is an equivalence relation.
Left Ideal is Left Module over Ring
Let $\struct {R, +, \times}$ be a ring. Let $J \subseteq R$ be a left ideal of $R$. Let $\circ : R \times J \to J$ be the restriction of $\times$ to $R \times J$. Then $\struct {J, +, \circ}$ is a left module over $\struct {R, +, \times}$.
Right Ideal is Right Module over Ring
Let $\struct {R, +, \times}$ be a ring. Let $J \subseteq R$ be a right ideal of $R$. Let $\circ : J \times R \to J$ be the restriction of $\times$ to $J \times R$. Then $\struct {J, +, \circ}$ is a right module over $\struct {R, +, \times}$.
Opposite Ring is Ring
Let $\struct {R, +, \times}$ be a ring. Let $\struct {R, +, *}$ be the opposite ring of $\struct {R, +, \times}$. Then $\struct {R, +, *}$ is a ring.
Opposite Ring of Opposite Ring
Let $\struct {R, +, \times}$ be a ring. Let $\struct {R, +, *}$ be the opposite ring of $\struct {R, +, \times}$. Let $\struct {R, +, \circ}$ be the opposite ring of $\struct {R, +, *}$. Then $\struct {R, +, \circ} = \struct {R, +, \times}$.
Left Module over Ring Induces Right Module over Opposite Ring
Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {R, +_R, *_R}$ be the opposite ring of $\struct {R, +_R, \times_R}$. Let $\struct{G, +_G, \circ}$ be a left module over $\struct {R, +_R, \times_R}$. Let $\circ’ : G \times R \to G$ be the binary operation defined by: :$\forall \lambda \in R: \forall x \in G: x \circ’ \lambda = \lambda \circ x$ Then $\struct{G, +_G, \circ’}$ is a right module over $\struct {R, +_R, *_R}$.
Right Module over Ring Induces Left Module over Opposite Ring
Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {R, +_R, *_R}$ be the opposite ring of $\struct {R, +_R, \times_R}$. Let $\struct{G, +_G, \circ}$ be a right module over $\struct {R, +_R, \times_R}$. Let $\circ’ : R \times G \to G$ be the binary operation defined by: :$\forall \lambda \in R: \forall x \in G: \lambda \circ’ x = x \circ \lambda $ Then $\struct{G, +_G, \circ’}$ is a left module over $\struct {R, +_R, *_R}$.
Ring is Commutative iff Opposite Ring is Itself
Let $\struct {R, +, \times}$ be a ring. Let $\struct {R, +, *}$ be the opposite ring of $\struct {R, +, \times}$. Then $\struct {R, +, \times}$ is a commutative ring : :$\struct {R, +, \times} = \struct {R, +, *}$
Left Module induces Right Module over same Ring iff Actions are Commutative
Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G, +_G, \circ}$ be a left module over $\struct {R, +_R, \times_R}$. Let $\circ': G \times R \to G$ be the binary operation defined by: :$\forall \lambda \in R: \forall x \in G: x \circ' \lambda = \lambda \circ x$ Then $\struct {G, +_G, \circ'}$ is a right module over $\struct {R, +_R, \times_R}$ : :$\forall \lambda, \mu \in R: \forall x \in G: \paren {\lambda \times_R \mu} \circ x = \paren {\mu \times_R \lambda} \circ x$
Right Module induces Left Module over same Ring iff Actions are Commutative
Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G, +_G, \circ}$ be a right module over $\struct {R, +_R, \times_R}$. Let $\circ' : R \times G \to G$ be the binary operation defined by: :$\forall \lambda \in R: \forall x \in G: \lambda \circ' x = x \circ \lambda $ Then $\struct {G, +_G, \circ'}$ is a left module over $\struct {R, +_R, \times_R}$ : :$\forall \lambda, \mu \in R: \forall x \in G: x \circ \paren{ \lambda \times_R \mu} = x \circ \paren {\mu \times_R \lambda}$
Euler's Theorem for Planar Graphs
Let $G = \struct {V, E}$ be a connected planar graph with $V$ vertices and $E$ edges. Let $F$ be the number of faces of $G$. Then: :$V - E + F = 2$
Fermat Problem
Let $\triangle ABC$ be a triangle Let the vertices of $\triangle ABC$ all have angles less than $120 \degrees$. Let $\triangle ABG$, $\triangle BCE$ and $\triangle ACF$ be equilateral triangles constructed on the sides of $ABC$. Let $AE$, $BF$ and $CG$ be constructed. Let $P$ be the point at which $AE$, $BF$ and $CG$ meet. :500px Then $P$ is the Fermat-Torricelli point of $\triangle ABC$. If one of vertices of $\triangle ABC$ be of $120 \degrees$ or more, then that vertex is itself the Fermat-Torricelli point of $\triangle ABC$.
Left Module over Commutative Ring induces Right Module
Let $\struct {R, +_R, \times_R}$ be a commutative ring. Let $\struct{G, +_G, \circ}$ be a left module over $\struct {R, +_R, \times_R}$. Let $\circ’ : G \times R \to G$ be the binary operation defined by: :$\forall \lambda \in R: \forall x \in G: x \circ’ \lambda = \lambda \circ x$ Then $\struct{G, +_G, \circ’}$ is a right module over $\struct {R, +_R, \times_R}$.
Right Module over Commutative Ring induces Left Module
Let $\struct {R, +_R, \times_R}$ be a commutative ring. Let $\struct{G, +_G, \circ}$ be a right module over $\struct {R, +_R, \times_R}$. Let $\circ' : R \times G \to G$ be the binary operation defined by: :$\forall \lambda \in R: \forall x \in G: \lambda \circ’ x = x \circ \lambda$ Then $\struct{G, +_G, \circ'}$ is a left module over $\struct {R, +_R, \times_R}$.
Right Ideal is Right Module over Ring/Ring is Right Module over Ring
Let $\struct {R, +, \times}$ be a ring. Then $\struct {R, +, \times}$ is a right module over $\struct {R, +, \times}$.
Right Module Does Not Necessarily Induce Left Module over Ring
Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct{G, +_G, \circ}$ be a right module over $\struct {R, +_R, \times_R}$. Let $\circ' : R \times G \to G$ be the binary operation defined by: :$\forall \lambda \in R: \forall x \in G: \lambda \circ' x = x \circ \lambda$ Then $\struct{G, +_G, \circ'}$ is not necessarily a left module over $\struct {R, +_R, \times_R}$
Left Module Does Not Necessarily Induce Right Module over Ring/Lemma
:$G$ is a left ideal of $\struct {\map {\MM_S} 2, +, \times}$.
Right Module Does Not Necessarily Induce Left Module over Ring/Lemma
:$G$ is a right ideal of $\struct {\map {\MM_S} 2, +, \times}$.
Ideal is Bimodule over Ring
Let $\struct {R, +, \times}$ be a ring. Let $J \subseteq R$ be an ideal of $R$. Let $\circ_l : R \times J \to J$ be the restriction of $\times$ to $R \times J$. Let $\circ_r : J \times R \to J$ be the restriction of $\times$ to $J \times R$. Then $\struct {J, +, \circ_l, \circ_r}$ is a bimodule over $\struct {R, +, \times}$.
Ideal is Bimodule over Ring/Ring is Bimodule over Ring
Let $\struct {R, +, \times}$ be a ring. Then $\struct {R, +, \times, \times}$ is a bimodule over $\struct {R, +, \times}$.
Left Module over Commutative Ring induces Bimodule
Let $\struct {R, +_R, \times_R}$ be a commutative ring. Let $\struct{G, +_G, \circ}$ be a left module over $\struct {R, +_R, \times_R}$. Let $\circ’ : G \times R \to G$ be the binary operation defined by: :$\forall \lambda \in R: \forall x \in G: x \circ’ \lambda = \lambda \circ x$ Then $\struct{G, +_G, \circ, \circ’}$ is a bimodule over $\struct {R, +_R, \times_R}$.
Right Module over Commutative Ring induces Bimodule
Let $\struct {R, +_R, \times_R}$ be a commutative ring. Let $\struct{G, +_G, \circ}$ be a right module over $\struct {R, +_R, \times_R}$. Let $\circ’ : R \times G \to G$ be the binary operation defined by: :$\forall \lambda \in R: \forall x \in G: \lambda \circ’ x = x \circ \lambda $ Then $\struct{G, +_G, \circ’, \circ}$ is a bimodule over $\struct {R, +_R, \times_R}$.
Diagonals of Kite are Perpendicular
Let $ABCD$ be a kite such that $AC$ and $BD$ are its diagonals. Then $AC$ and $BD$ are perpendicular.
One Diagonal of Kite Bisects the Other
Let $ABCD$ be a kite such that: :$AC$ and $BD$ are its diagonals :$AB = BC$ :$AD = DC$ Then $BD$ is the perpendicular bisector of $AC$.
Theorem of Even Perfect Numbers/Historical Note
The first part of this proof of the '''Theorem of Even Perfect Numbers''' was documented by {{AuthorRef|Euclid}} in {{BookLink|The Elements|Euclid}}: {{EuclidPropLink|book = IX|prop = 36|title = Theorem of Even Perfect Numbers/Sufficient Condition}}. The second part was achieved by {{AuthorRef|Leonhard Paul Euler|Euler}}.
Combination Theorem for Continuous Mappings/Topological Group
Let $\struct{S, \tau_{_S}}$ be a topological space. Let $\struct{G, *, \tau_{_G}}$ be a topological group. Let $\lambda \in G$. Let $f,g : \struct{S, \tau_{_S}} \to \struct{G, \tau_{_G}}$ be continuous mappings. Then the following results hold: === Product Rule === {{:Combination Theorem for Continuous Mappings/Topological Group/Product Rule}} === Multiple Rule === {{:Combination Theorem for Continuous Mappings/Topological Group/Multiple Rule}} === Inverse Rule === {{:Combination Theorem for Continuous Mappings/Topological Group/Inverse Rule}}
Combination Theorem for Continuous Mappings/Topological Group/Product Rule
:$f * g : \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping.
Combination Theorem for Continuous Mappings/Topological Group/Multiple Rule
:$\lambda * f: \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping :$f * \lambda: \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping.
Combination Theorem for Continuous Mappings/Topological Group/Inverse Rule
:$g^{-1}: \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping.
Continuous Mapping to Topological Product/Corollary
Let $T = T_1 \times T_2$ be a product space of two topological spaces $T_1$ and $T_2$. Let $T'$ be a topological space. Let $f: T' \to T_1$ be a mapping. Let $g: T' \to T_2$ be a mapping. Let $f \times g : T’ \to T$ be the mapping defined by: :$\forall x \in T’ : \map {\paren {f \times g}} x = \tuple{ \map f x, \map g x}$ Then $f \times g$ is continuous  $f$ and $g$ are continuous.