hcju/proofwikips huggingface dataset:
Finite Integral Domain cannot be Ordered
Let $\struct {D, +, \times}$ be an integral domain whose unity is $1_D$ and whose order is finite. Then $D$ cannot be an ordered integral domain.
Ring of Integers Modulo m cannot be Ordered Integral Domain
Let $m \in \Z: m \ge 2$. Let $\struct {\Z_m, +, \times}$ be the ring of integers modulo $m$. Then $\struct {\Z_m, +, \times}$ cannot be an ordered integral domain.
Strict Positivity Property induces Total Ordering
Let $\struct {D, +, \times}$ be an integral domain whose zero is $0_D$. Let $D$ be endowed with a (strict) positivity property $P: D \to \set {\mathrm T, \mathrm F}$. Then there exists a total ordering $\le$ on $\struct {D, +, \times}$ induced by $P$ which is compatible with the ring structure of $\struct {D, +, \times}$.
Relation Induced by Strict Positivity Property is Transitive
Let $\struct {D, +, \times}$ be an ordered integral domain where $P$ is the (strict) positivity property. Let the relation $<$ be defined on $D$ as: :$\forall a, b \in D: a < b \iff \map P {-a + b}$ Then $<$ is a transitive relation.
Relation Induced by Strict Positivity Property is Compatible with Addition
Let $\struct {D, +, \times}$ be an ordered integral domain where $P$ is the (strict) positivity property. Let the relation $<$ be defined on $D$ as: :$\forall a, b \in D: a < b \iff \map P {-a + b}$ Then $<$ is compatible with $+$, that is: :$\forall x, y, z \in D: x < y \implies \paren {x + z} < \paren {y + z}$ :$\forall x, y, z \in D: x < y \implies \paren {z + x} < \paren {z + y}$
Relation Induced by Strict Positivity Property is Compatible with Multiplication
Let $\struct {D, +, \times}$ be an ordered integral domain where $P$ is the (strict) positivity property. Let the relation $<$ be defined on $D$ as: :$\forall a, b \in D: a < b \iff \map P {-a + b}$ Then $<$ is compatible with $\times$ in the following sense: :$\forall x, y, z \in D: x < y, \map P z \implies \paren {z \times x} < \paren {z \times y}$ :$\forall x, y, z \in D: x < y, \map P z \implies \paren {x \times z} < \paren {y \times z}$
Relation Induced by Strict Positivity Property is Trichotomy
Let $\struct {D, +, \times}$ be an ordered integral domain where $P$ is the (strict) positivity property. Let the relation $<$ be defined on $D$ as: :$\forall a, b \in D: a < b \iff \map P {-a + b}$ Then $\forall a, b \in D:$ exactly one of the following conditions applies: :$a < b$ :$a = b$ :$a > b$ That is, $<$ is a trichotomy.
Relation Induced by Strict Positivity Property is Asymmetric and Antireflexive
Let $\struct {D, +, \times}$ be an ordered integral domain where $P$ is the (strict) positivity property. Let the relation $<$ be defined on $D$ as: :$\forall a, b \in D: a < b \iff \map P {-a + b}$ Then $<$ is asymmetric and antireflexive.
Properties of Strict Negativity
Let $\struct {D, +, \times}$ be an ordered integral domain, whose (strict) positivity property is denoted $P$. Let $\le$ be the total ordering induced by $P$, and let $<$ be its strict total ordering counterpart. Let $N$ be the (strict) negativity property on $D$: :$\forall a \in D: \map N a \iff \map P {-a}$ Then the following properties apply for all $a, b \in D$:   === Strict Negativity is equivalent to Strictly Preceding Zero === {{:Strict Negativity is equivalent to Strictly Preceding Zero}}
Sum of Absolute Values on Ordered Integral Domain
Let $\struct {D, +, \times, \le}$ be an ordered integral domain. For all $a \in D$, let $\size a$ denote the absolute value of $a$. Then: :$\size {a + b} \le \size a + \size b$
One Succeeds Zero in Well-Ordered Integral Domain
Let $\struct {D, +, \times, \le}$ be a well-ordered integral domain. Let $0$ and $1$ be the zero and unity respectively of $D$. Then $0$ is the immediate predecessor of $1$: :$0 < 1$ :$\neg \exists a \in D: 0 < a < 1$
Square of Element Less than Unity in Ordered Integral Domain
Let $\struct {D, +, \times, \le}$ be an ordered integral domain. Let $x \in D$ such that $0 < x < 1$. Then: : $0 < x \times x < x$
Principle of Mathematical Induction/Well-Ordered Integral Domain
Let $\struct {D, +, \times, \le}$ be a well-ordered integral domain whose zero is $0_D$. Let the unity of $D$ be $1_D$. Let $S \subseteq D$ be such that: :$1_D \in S$ :$a \in S \implies a + 1_D \in S$ Then: :$D_{> 0_D} \subseteq S$ where $D_{> 0_D}$ denotes all the elements $d \in D$ such that $\map P d$. That is, $D_{> 0_D}$ is the set of all (strictly) positive elements of $D$.
Multiple of Divisor in Integral Domain Divides Multiple
Let $\struct {D, +, \times}$ be an integral domain. Let $a, b, c \in D$. Let $a \divides b$, where $\divides$ denotes divisibility. Then $a \times c$ is a divisor of $b \times c$.
Integer Divisor Results/One Divides all Integers
{{begin-eqn}} {{eqn | l = 1       | o = \divides       | r = n }} {{eqn | l = -1       | o = \divides       | r = n }} {{end-eqn}}
Integer Divisor Results/Integer Divides Itself
:$n \divides n$
Integer Divisor Results/Integer Divides its Negative
{{begin-eqn}} {{eqn | l = n       | o = \divides       | r = -n }} {{eqn | l = -n       | o = \divides       | r = n }} {{end-eqn}}
Integer Divisor Results/Integer Divides its Absolute Value
{{begin-eqn}} {{eqn | l = n       | o = \divides       | r = \size n }} {{eqn | l = \size n       | o = \divides       | r = n }} {{end-eqn}} where: :$\size n$ is the absolute value of $n$ :$\divides$ denotes divisibility.
Integer Divisor Results/Integer Divides Zero
:$n \divides 0$
Integer Divisor Results/Divisors of Negative Values
: $m \mathrel \backslash n \iff -m \divides n \iff m \divides -n \iff -m \divides -n$
Divisors of One
The only divisors of $1$ are $1$ and $-1$. That is: :$a \divides 1 \iff a = \pm 1$
Divisor Relation is Antisymmetric
''Divides'' is a antisymmetric relation on $\Z_{>0}$, the set of positive integers. That is: :$\forall a, b \in \Z_{>0}: a \divides b \land b \divides a \implies a = b$
Real Function is Continuous at Point iff Oscillation is Zero
Let $f: D \to \R$ be a real function where $D \subseteq \R$. Let $x$ be a point in $D$. Let $N_x$ be the set of open subset neighborhoods of $x$. Let $\map {\omega_f} x$ be the oscillation of $f$ at $x$: :$\map {\omega_f} x = \displaystyle \inf \set {\map {\omega_f} I: I \in N_x}$ where: :$\map {\omega_f} I = \displaystyle \sup \set {\size {\map f y - \map f z}: y, z \in I \cap D}$ Then $\map {\omega_f} x = 0$  $f$ is continuous at $x$.
Numbers of Type Integer a plus b root 2 are Not a Field
Let $\Z \sqbrk {\sqrt 2}$ denote the set: :$\Z \sqbrk {\sqrt 2} := \set {a + b \sqrt 2: a, b \in \Z}$ that is, all (real) numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are integers. Then the algebraic structure: :$\struct {\Z \sqbrk {\sqrt 2}, +, \times}$ where $+$ and $\times$ are addition and multiplication on real numbers, is '''not''' a field.
Numbers of Type Rational a plus b root 2 form Field
Let $\Q \sqbrk {\sqrt 2}$ denote the set: :$\Q \sqbrk {\sqrt 2} := \set {a + b \sqrt 2: a, b \in \Q}$ that is, all numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are rational numbers. Then the algebraic structure: :$\struct {\Q \sqbrk {\sqrt 2}, +, \times}$ where $+$ and $\times$ are conventional addition and multiplication on real numbers, is a field.
Characteristic of Ring of Integers Modulo Prime
Let $\struct {\Z_p, +, \times}$ be the ring of integers modulo $p$, where $p$ is a prime number. The characteristic of $\struct {\Z_p, +, \times}$ is $p$.
Gamma Function Extends Factorial
:$\forall n \in \N: \map \Gamma {n + 1} = n!$
Uniform Limit of Analytic Functions is Analytic
Let $U$ be an open subset of $\C$. Let $\left\{ {f_n}\right\}_{n \mathop \in \N}$ be a sequence of analytic functions $f_n : U \to \C$. Let $\left\{ {f_n}\right\}$ converge locally uniformly to $f$ on $U$. Then $f$ is analytic.
Modulus of Complex Integral
Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval. Let $f : \left[{a \,.\,.\, b}\right] \to \C$ be a continuous complex function. Then: :$\displaystyle \left|{\int_a^b f \left({t}\right) \rd t}\right| \le \int_a^b \left|{f \left({t}\right)}\right| \rd t$ where the first integral is a complex Riemann integral, and the second integral is a definite real integral.
Triangle Inequality for Contour Integrals
Let $C$ be a contour. Let $f: \Img C \to \C$ be a continuous complex function, where $\Img C$ denotes the image of $C$. Then: :$\displaystyle \size {\int_C \map f z \rd z} \le \max_{z \mathop \in \Img C} \size {\map f z} \map L C$ where $\map L C$ denotes the length of $C$.
Jensen's Formula
Let $S$ be an open subset of the complex plane containing the closed disk: :$D_r = \set {z \in \C : \cmod z \le r}$ of radius $r$ about $0$. Let $f: S \to \C$ be holomorphic on $S$.  Let $f$ have no zeroes on the circle $\cmod z = r$. Let $\map f 0 \ne 0$. Let $\rho_1, \ldots, \rho_n$ be the zeroes of $f$ in $D_r$, counted with multiplicity. Then: :$(1): \quad \displaystyle \frac 1 {2 \pi} \int_0^{2 \pi} \ln \cmod {\map f {r e^{i \theta} } } \rd \theta = \ln \cmod {\map f 0} + \sum_{k \mathop = 1}^n \paren {\ln r - \ln \size {\rho_k} }$
Residue Theorem
Let $U$ be a simply connected open subset of the complex plane $\C$. Let $a_1, a_2, \dots, a_n$ be finitely many points of $U$. Let $f: U \to \C$ be analytic in $U \setminus \set {a_1, a_2, \dots, a_n}$.  Let $L = \partial U$ be oriented counterclockwise. Then: :$\displaystyle \oint_L \map f z \rd z = 2 \pi i \sum_{k \mathop = 1}^n \Res f {a_k}$
Cauchy's Integral Formula
Let $D = \set {z \in \C: \cmod z \le r}$ be the closed disk of radius $r$ in $\C$. Let $f: U \to \C$ be holomorphic on some open set containing $D$. Then for each $a$ in the interior of $D$: :$\displaystyle \map f a = \frac 1 {2 \pi i} \int_{\partial D} \frac {\map f z} {\paren {z - a} } \rd z$ where $\partial D$ is the boundary of $D$, and is traversed anticlockwise.
Cauchy's Integral Formula/General Result
Let $n \in \N$ be a natural number. Then for each $a$ in the interior of $D$: :$\displaystyle f^{\paren n} \paren a = \dfrac {n!} {2 \pi i} \int_{\partial D} \frac {\map f z} {\paren {z - a}^{n + 1} } \rd z$ where $\partial D$ is the boundary of $D$, and is traversed anticlockwise.
Characteristic of Subfield of Complex Numbers is Zero
The characteristic of any subfield of the field of complex numbers is $0$.
Characteristic times Ring Element is Ring Zero
Let $\struct {R, +, \circ}$ be a ring with unity. Let the zero of $R$ be $0_R$ and the unity of $R$ be $1_R$. Let the characteristic of $R$ be $n$. Then: :$\forall a \in R: n \cdot a = 0_R$
Field of Rational Functions is Field
Let $K$ be a field. Let $K \sqbrk x$ be the integral domain of polynomial forms on $K$. Let $\map K x$ be the field of rational functions on $K$. Then $\map K x$ forms a field. If the characteristic of $K$ is $p$, then the characteristic of $\map K x$ is non-zero.
Hadamard Factorization Theorem
Let $f: \C \to \C$ be an entire function of finite order $\omega$. Let $0$ be a zero of $f$ of multiplicity $m\geq0$. Let $\left\langle{a_n}\right\rangle$ be the sequence of nonzero zeroes of $f$, repeated according to multiplicity. Then $f$ has finite rank $p\leq\omega$ and there exists a polynomial $g$ of degree at most $\omega$ such that: :$\displaystyle f \left({z}\right) = z^m e^{g(z)} \prod_{n \mathop = 1}^\infty E_p\left( \frac z{a_n} \right)$ where $E_p$ denotes the $p$th Weierstrass elementary factor.
Properties of Gamma Function
The gamma function $\Gamma \left({z}\right)$ has the following properties:
Poisson Summation Formula
Let $f: \R \to \C$ be a Schwarz function. Let $\hat f$ be its Fourier transform. Then: :$\displaystyle \sum_{n \mathop \in \Z} \map f n = \sum_{m \mathop \in \Z} \map {\hat f} m$
Harmonic Properties of Schwarz Functions
Let $f, g : \R \to \C$ be Schwarz functions. Let $\hat f$, $\hat g$ be the Fourier transforms of $f$ and $g$ respectively. Then: :$(1): \quad \hat f$, $\hat g$ are Schwarz functions. :$(2): \quad \map {\widehat {\paren {\hat f} } } x = \map f {-x}$ for all $x \in \R$. :$(3): \quad$ If $f * g$ is the convolution of $f$ and $g$, then: {{disambiguate|Definition:Convolution}} ::::$\widehat {f * g} = \hat f \hat g$
Euler's Reflection Formula
:$\forall z \notin \Z: \map \Gamma z \, \map \Gamma {1 - z} = \dfrac \pi {\map \sin {\pi z} }$
Legendre's Duplication Formula
:$\forall z \notin \set {-\dfrac n 2: n \in \N}: \map \Gamma z \, \map \Gamma {z + \dfrac 1 2} = 2^{1 - 2 z} \sqrt \pi \, \map \Gamma {2 z}$ where $\N$ denotes the natural numbers.
Approximation to Reciprocal times Derivative of Gamma Function
For all $z \in \C$ such that $\cmod {\map \arg z} < \pi - \epsilon, \cmod z > 1$: :$\dfrac {\map {\Gamma'} z} {\map \Gamma z} = \ln z + \map {\mathcal O_\epsilon} {z^{-1} }$ where: :$\map {\mathcal O} {z^{-1} }$ denotes big-O notation :the implied constant depends on $\epsilon$.
Complex Conjugate of Gamma Function
:$\forall z \in \C \setminus \set {0, -1, -2, \ldots}: \map \Gamma {\overline z} = \overline {\map \Gamma z}$
Intersection of Integer Ideals is Lowest Common Multiple
Let $\ideal m$ and $\ideal n$ be ideals of the integers $\Z$. Let $\ideal k$ be the intersection of $\ideal m$ and $\ideal n$. Then $k = \lcm \set {m, n}$.
LCM iff Divides All Common Multiples
Let $a, b \in \Z$ such that $a b \ne 0$. Let $m \in \Z: d > 0$. Then $m = \lcm \set {a, b}$ : : $(1): \quad a \divides m \land b \divides m$ : $(2): \quad a \divides n \land b \divides n \implies m \divides n$ That is, in the set of positive integers, $m$ is the LCM of $a$ and $b$  $m$ is a common multiple of $a$ and $b$, and $m$ also divides any other common multiple of $a$ and $b$.
LCM Divides Common Multiple
Let $a, b \in \Z$ such that $a b \ne 0$. Let $n$ be any common multiple of $a$ and $b$. That is, let $n \in \Z: a \divides n, b \divides n$. Then: :$\lcm \set {a, b} \divides n$ where $\lcm \set {a, b}$ is the lowest common multiple of $a$ and $b$. {{:Euclid:Proposition/VII/35}}
Sum of Integer Ideals is Greatest Common Divisor
Let $\ideal m$ and $\ideal n$ be ideals of the integers $\Z$. Let $\ideal d = \ideal m + \ideal n$. Then $d = \gcd \set {m, n}$.
Quotient Ring of Integers with Principal Ideal
Let $\struct {\Z, +, \times}$ be the integral domain of integers. Let $n \in \Z$. Let $\ideal n$ be the principal ideal of $\struct {\Z, +, \times}$ generated by $n$. The quotient ring $\struct {\Z, +, \times} / \ideal n$ is isomorphic to $\struct {\Z_n, +_n, \times_n}$, the ring of integers modulo $n$. Note the special cases where $n = 0$ or $1$: :Quotient Ring of Integers and Zero :Quotient Ring of Integers and Principal Ideal from Unity
Quotient Ring by Null Ideal
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $\struct {\set {0_R}, +, \circ}$ be the null ideal of $\struct {R, +, \circ}$. Let $\struct {R / \set {0_R}, +, \circ}$ be the quotient ring of $R$ defined by $\set {0_R}$. Then $\struct {R / \set {0_R}, +, \circ}$ is isomorphic to $\struct {R, +, \circ}$.
Ring Homomorphism Preserves Negatives
Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring homomorphism. Then: :$\forall x \in R_1: \map \phi {-x} = -\paren {\map \phi x}$
Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function
Let $\zeta$ be the Riemann zeta function. Let $s \in \C$ be a complex number with real part $\sigma>1$. Then: :$\displaystyle \pi^{-s / 2} \Gamma \left({\frac s 2}\right) \zeta \left({s}\right) = - \frac 1 {s \left({1 - s}\right)} + \int_1^\infty \left({x^{s / 2 - 1} + x^{- \left({s + 1}\right) / 2} }\right) \omega \left({x}\right) \rd x$ where: : $\Gamma$ is the gamma function : $\displaystyle \omega \left({x}\right) = \sum_{n \mathop = 1}^\infty e^{- \pi n^2 x}$
Mapping on Integers a plus b root 2 to Conjugate is Automorphism
Let $\Z \sqbrk {\sqrt 2}$ denote the set: :$\Z \sqbrk {\sqrt 2} := \set {a + b \sqrt 2: a, b \in \Z}$ that is, all numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are integers. Then the mapping $\phi: \Z \sqbrk {\sqrt 2} \to \Z \sqbrk {\sqrt 2}$ defined as: :$\forall x = a + b \sqrt 2 \in \Z \sqbrk {\sqrt 2}: \map \phi {a + b \sqrt 2} = a - b \sqrt 2$ is a ring automorphism.
Ring Epimorphism from Integers to Integers Modulo m
Let $\struct {\Z, +, \times}$ be the ring of integers. Let $\struct {\Z_m, +_m, \times_m}$ be the ring of integers modulo $m$. Let $\phi: \struct {\Z, +, \times} \to \struct {\Z_m, +_m, \times_m}$ be the mapping defined as: :$\forall x \in \Z: \map \phi x = \eqclass x m$ where $\eqclass x m$ is the residue class modulo $m$. Then $\phi$ is a ring epimorphism, but specifically not a ring monomorphism. The image of $\phi$ is $\struct {\Z_m, +_m, \times_m}$. The kernel of $\phi$ is $m \Z$, the set of integer multiples of $m$.
Monotone Additive Function is Linear
Let $f: \R \to \R$ be a monotone real function which is additive, that is: :$\forall x, y \in \R: \map f {x + y} = \map f x + \map f y$ Then: :$\exists a \in \R: \forall x \in \R: \map f x = a x$
Units of Ring of Polynomial Forms over Field
Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $F \sqbrk X$ be the ring of polynomial forms in an indeterminate $X$ over $F$. Then the units of $F \sqbrk X$ are all the elements of $F \sqbrk X$ whose degree is $0$.
Units of Numbers of Type Integer a plus b root 2
Let $\Z \sqbrk {\sqrt 2}$ denote the set: :$\Z \sqbrk {\sqrt 2} := \set {a + b \sqrt 2: a, b \in \Z}$ that is, all numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are integers. Let $\struct {\Z \sqbrk {\sqrt 2}, +, \times}$ be the integral domain where $+$ and $\times$ are conventional addition and multiplication on real numbers. Then numbers of the form $a + b \sqrt 2$ such that $a^2 - 2 b^2 = \pm 1$ are all units of $\struct {\Z \sqbrk {\sqrt 2}, +, \times}$.
Units of Gaussian Integers
Let $\struct {\Z \sqbrk i, +, \times}$ be the ring of Gaussian integers. The set of units of $\struct {\Z \sqbrk i, +, \times}$ is $\set {1, i, -1, -i}$.
Integers are Euclidean Domain
The integers $\Z$ with the mapping $\nu: \Z \to \Z$ defined as: :$\forall x \in \Z: \map \nu x = \size x$ form a Euclidean domain.
Gaussian Integers form Euclidean Domain
Let $\struct {\Z \sqbrk i, +, \times}$ be the integral domain of Gaussian Integers. Let $\nu: \Z \sqbrk i \to \R$ be the real-valued function defined as:  :$\forall a \in \Z \sqbrk i: \map \nu a = \cmod a^2$  where $\cmod a$ is the (complex) modulus of $a$. Then $\nu$ is a Euclidean valuation on $\Z \sqbrk i$. Hence $\struct {\Z \sqbrk i, +, \times}$ with $\nu: \Z \sqbrk i \to \Z$ forms a Euclidean domain.
Degree of Product of Polynomials over Ring
:$\forall f, g \in R \left[{X}\right]: \deg \left({f g}\right) \le \deg \left({f}\right) + \deg \left({g}\right)$
Degree of Sum of Polynomials
:$\forall f, g \in R \sqbrk X: \map \deg {f + g} \le \max \set {\map \deg f, \map \deg g}$
Degree of Product of Polynomials over Integral Domain not Less than Degree of Factors
:$\forall f, g \in R \sqbrk X: \map \deg {f g} \ge \map \deg f$
Polynomial Forms over Field is Euclidean Domain
Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $X$ be transcendental in $F$. Let $F \sqbrk X$ be the ring of polynomial forms in $X$ over $F$. Then $F \sqbrk X$ is a Euclidean domain.
Euclidean Domain is Principal Ideal Domain
A Euclidean domain is a principal ideal domain.
Elements of Euclidean Domain have Greatest Common Divisor
Let $\struct {D, +, \times}$ be a Euclidean domain. Then any two elements $a, b \in D$ have a greatest common divisor $d$ such that: :$d \divides a \land d \divides b$ :$x \divides a \land x \divides b \implies x \divides d$ and $d$ is written $\gcd \set {a, b}$. For any $a, b \in D$: :$\exists s, t \in D: s a + t b = d$ Any two greatest common divisors of any $a, b$ are associates.
Associates in Ring of Polynomial Forms over Field
Let $F \sqbrk X$ be the ring of polynomial forms over the field $F$. Let $\map d X$ and $\map {d'} X$ be polynomial forms in $F \sqbrk X$. Then $\map d X$ is an associate of $\map {d'} X$  $\map d X = c \cdot \map {d'} X$ for some $c \in F, c \ne 0_F$. Hence any two polynomials in $F \sqbrk X$ have a unique monic GCD.
Zero Element Generates Null Ideal
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. For $r \in R$, let $\ideal r$ denote the ideal generated by $r$. Then $\ideal {0_R}$ is the null ideal.
Irreducible Elements of Ring of Integers
Let $\struct {\Z, +, \times}$ be the ring of integers. The irreducible elements of $\struct {\Z, +, \times}$ are the prime numbers and their negatives.
Euclid's Lemma for Euclidean Domains
Let $\struct {D, +, \times}$ be a Euclidean domain whose unity is $1$. Let $a, b, c \in D$. Let $a \divides b \times c$, where $\divides$ denotes divisibility. Let $a \perp b$, where $\perp$ denotes relative primeness. Then $a \divides c$.
Rational Polynomial is Content Times Primitive Polynomial
Let $\Q \sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $\map f X \in \Q \sqbrk X$. Then: :$\map f X = \cont f \, \map {f^*} X$ where: :$\cont f$ is the content of $\map f X$ :$\map {f^*} X$ is a primitive polynomial. For a given polynomial $\map f X$, both $\cont f$ and $\map {f^*} X$ are unique.
Product of Rational Polynomials
Let $\Q \sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $\map f X, \map g X \in \Q \sqbrk X$. Using Rational Polynomial is Content Times Primitive Polynomial, let these be expressed as: :$\map f X = \cont f \cdot \map {f^*} X$ :$\map g X = \cont g \cdot \map {g^*} X$ where: :$\cont f, \cont g$ are the content of $f$ and $g$ respectively :$f^*, g^*$ are primitive. Let $\map h X = \map f X \, \map g X$ be the product of $f$ and $g$. Then: :$\map {h^*} X = \map {f^*} X \, \map {g^*} X$
Parallelogram Law
Let $\mathbf a$ and $\mathbf b$ be vector quantities. Consider a parallelogram, two of whose adjacent sides represent $\mathbf a$ and $\mathbf b$ (in magnitude and direction). :300px Then the diagonal of the parallelogram through that common point represents the magnitude and direction of $\mathbf a + \mathbf b$, the sum of $\mathbf a$ and $\mathbf b$.
Equivalence of Definitions of Perfect Set
{{TFAE|def = Perfect Set}} Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a subset of $S$.
Complement of Interior equals Closure of Complement
Let $T$ be a topological space. Let $H \subseteq T$. Let $H^-$ denote the closure of $H$ and $H^\circ$ denote the interior of $H$. Let $\map \complement H$ be the complement of $H$ in $T$: :$\map \complement H = T \setminus H$ Then: :$\map \complement {H^\circ} = \paren {\map \complement H}^-$ and similarly: :$\paren {\map \complement H}^\circ = \map \complement {H^-}$ These can alternatively be written: :$T \setminus H^\circ = \paren {T \setminus H}^-$ :$\paren {T \setminus H}^\circ = T \setminus H^-$ which, it can be argued, is easier to follow.
Interior of Finite Intersection equals Intersection of Interiors
Let $T$ be a topological space. Let $n \in \N$. Let: :$\forall i \in \set {1, 2, \dotsc, n}: H_i \subseteq T$ Then: :$\displaystyle \left({\bigcap_{i \mathop = 1}^n H_i}\right)^\circ = \bigcap_{i \mathop = 1}^n H_i^\circ$ where $H_i^\circ$ denotes the interior of $H_i$.
Finite Union of Regular Closed Sets is Regular Closed
Let $T$ be a topological space. Let $n \in \N$. Suppose that: :$\forall i \in \set {1, 2, \dotsc, n}: H_i \subseteq T$ where all the $H_i$ are regular closed in $T$. That is: :$\forall i \in \set {1, 2, \dotsc, n}: H_i = H_i^{\circ -}$ where $H_i^{\circ -}$ denotes the closure of the interior of $H_i$ Then $\displaystyle \bigcup_{i \mathop = 1}^n H_i$ is regular closed in $T$. That is: :$\displaystyle \bigcup_{i \mathop = 1}^n H_i = \paren {\bigcup_{i \mathop = 1}^n H_i}^{\circ -}$
Boundary is Intersection of Closure with Closure of Complement
Let $T$ be a topological space. Let $X \subseteq T$. Let $\partial X$ denote the boundary of $X$, defined as: :$\partial X = X^- \setminus X^\circ$ Let $\overline X = T \setminus X$ denote the complement of $X$ in $T$. Let $X^-$ denote the closure of $X$. Then: :$\partial X = X^- \cap \paren {\overline X}^-$
Set is Closed iff it Contains its Boundary
Let $T$ be a topological space, and let $H \subseteq T$. Then $H$ is closed in $T$ : :$\partial H \subseteq H$ where $\partial H$ is the boundary of $H$.