hcju/proofwikips huggingface dataset:
Bounded Piecewise Continuous Function may not have One-Sided Limits
Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$, $a < b$.  Let $f$ be a bounded piecewise continuous function. {{:Definition:Bounded Piecewise Continuous Function}} Then it is not necessarily the case that $f$ is a piecewise continuous function with one-sided limits: {{:Definition:Piecewise Continuous Function with One-Sided Limits}}
Linear Second Order ODE/y'' = 1 over 1 - x^2
The second order ODE: :$(1): \quad y'' = \dfrac 1 {1 - x^2}$ has the general solution: :$y = x \tanh^{-1} x + \map \ln {1 - x^2} + C x + D$
Universal Property of Polynomial Ring/Free Monoid on Set
Let $R, S$ be commutative and unitary rings. Let $\left\langle{s_j}\right\rangle_{j \mathop \in J}$ be an indexed family of elements of $S$. Let $\psi: R \to S$ be a ring homomorphism. Let $R \left[{\left\{{X_j: j \in J}\right\}}\right]$ be a polynomial ring. Then there exists a unique evaluation homomorphism $\phi: R \left[{\left\{{X_j: j \in J}\right\}}\right] \to S$ at $\langle s_j\rangle_{j \in J}$ extending $\psi$.
Polynomial Addition is Associative
Addition of polynomials is an associative operation.
Null Polynomial is Additive Identity
The set of polynomial forms has an additive identity. {{explain|Context}}
De Morgan's Laws (Set Theory)/Proof by Induction/Difference with Union/Proof
Let $\mathbb T = \set {T_i: i \mathop \in I}$, where each $T_i$ is a set and $I$ is some finite indexing set. Then: : $\displaystyle S \setminus \bigcup_{i \mathop \in I} T_i = \bigcap_{i \mathop \in I} \paren {S \setminus T_i}$
Parity Addition is Commutative
Let $R := \struct {\set {\text{even}, \text{odd} }, +, \times}$ be the parity ring. The operation $+$ is commutative: :$\forall a, b \in R: a + b = b + a$
Parity Multiplication is Associative
Let $R := \struct {\set {\text{even}, \text{odd} }, +, \times}$ be the parity ring. The operation $\times$ is associative: :$\forall a, b, c \in R: \paren {a \times b} \times c = a \times \paren {b \times c}$
Events One of Which equals Union
Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$. Let $A, B \in \Sigma$ be events of $\EE$, so that $A \subseteq \Omega$ and $B \subseteq \Omega$. Let $A$ and $B$ be such that: :$A \cup B = A$ Then whenever $B$ occurs, it is always the case that $A$ occurs as well.
Direct Product iff Nontrivial Idempotent
Let $R$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Then $R$ is the direct product of two non-trivial rings  $R$ contains an idempotent element not equal to $0_R$ or $1_R$.
Real Numbers under Multiplication do not form Group
The algebraic structure $\struct {\R, \times}$ consisting of the set of real numbers $\R$ under multiplication $\times$ is not a group.
Total Variation is Non-Negative
Let $a, b$ be real numbers with $a < b$. Let $f : \closedint a b \to \R$ be a function of bounded variation. Let $V_f$ be the total variation of $f$ on $\closedint a b$. Then:  :$V_f \ge 0$ with equality  $f$ is constant.
Numbers of Type Integer a plus b root 2 Form Ordered Integral Domain
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 algebraic structure: :$\struct {\Z \sqbrk {\sqrt 2}, +, \times}$ where $+$ and $\times$ are conventional addition and multiplication on real numbers, is an ordered integral domain.
Basis Test for Isolated Point
Let $T = \struct {S, \tau}$ be a topological space. Let $\BB$ be a synthetic basis of $T$. Let $H \subseteq S$. Then $x \in H$ is an isolated point of $H$ : :$\exists U \in \BB : U \cap H = \set x$
Boundary of Boundary is not necessarily Equal to Boundary
Let $T$ be a topological space. Let $H \subseteq T$. Let $\partial H$ denote the boundary of $H$. While it is true that: :$\map \partial {\partial H} \subseteq \partial H$ it is not necessarily the case that: :$\map \partial {\partial H} = \partial H$
Right Congruence Class Modulo Subgroup is Right Coset
Let $\mathcal R^r_H$ be the equivalence defined as right congruence modulo $H$. The equivalence class $\eqclass g {\mathcal R^r_H}$ of an element $g \in G$ is the right coset $H g$. This is known as the '''right congruence class of $g \bmod H$'''.
Equivalence of Definitions of Filter Basis
{{TFAE|def = Filter Basis}} Let $S$ be a set. Let $\FF$ be a filter on $S$.
Lexicographic Order on Pair of Well-Ordered Sets is Well-Ordering
Let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be ordered sets. Let $\preccurlyeq$ be the lexicographic order on $S_1 \times S_2$''': :$\tuple {x_1, x_2} \preccurlyeq \tuple {y_1, y_2} \iff \tuple {x_1 \prec_1 y_1} \lor \tuple {x_1 = y_1 \land x_2 \preceq_2 y_2}$ Then: :$\preccurlyeq$ is a well-ordering on $S_1 \times S_2$  :both $\preceq_1$ and $\preceq_2$ are well-orderings.
Product of Functions of Bounded Variation is of Bounded Variation
Let $a, b$ be real numbers with $a < b$. Let $f, g : \closedint a b \to \R$ be functions of bounded variation. Let $V_f$ and $V_g$ be the total variations of $f$ and $g$ respectively.  Then $f \times g$ is of bounded variation with: :$V_{f \times g} \le A V_f + B V_g$ where: :$V_{f \times g}$ denotes the total variation of $f \times g$ :$A, B$ are non-negative real numbers.
Difference of Functions of Bounded Variation is of Bounded Variation
Let $a, b$ be real numbers with $a < b$. Let $f, g : \closedint a b \to \R$ be functions of bounded variation. Let $V_f$ and $V_g$ be the total variations of $f$ and $g$ respectively.  Then $f - g$ is of bounded variation with: :$V_{f - g} \le V_f + V_g$ where $V_{f - g}$ denotes the total variation of $f - g$.
Approximation to Stirling's Formula for Gamma Function
Let: :$D_\epsilon = \left\{{z \in \C : \left\vert{\operatorname{Arg} \left({z}\right)}\right\vert < \pi - \epsilon,\ \left\vert{z}\right\vert > 1}\right\}$ where: :$\left\vert{\operatorname{Arg} \left({z}\right)}\right\vert$ denotes the absolute value of the principal argument of $z$ :$\left\vert{z}\right\vert$ denotes the modulus of $z$ :$\epsilon \in \R_{>0}$. Then for all $z \in D_\epsilon$, the gamma function of $z$ satisfies: :$\Gamma \left({z}\right) = \sqrt{\dfrac {2 \pi} z} \left({\dfrac z e}\right)^z\left({1 + \mathcal O \left({z^{-1} }\right)}\right)$ where $\mathcal O \left({z^{-1} }\right)$ denotes big-O of $z^{-1}$.
Cauchy Sequence in Metric Space is not necessarily Convergent
Let $M = \struct {A, d}$ be a metric space. Let $\sequence {x_n}$ be a Cauchy sequence in $M$. Then it is not necessarily the case that $M$ is a convergent sequence in $M$.
Integers are not Close Packed
The integers $\Z$ are not close packed. That is: :$\forall n \in \Z: \not \exists m \in \Z: n < m < n + 1$
Linear Combination of Integers is Ideal
Let $a, b$ be any integers. Let $\Bbb S = \set {a x + b y: x, y \in \Z}$. Then the algebraic structure: :$\struct {\Bbb S, +, \times}$ is an ideal of $\Z$.
Negative of Sum of Real Numbers/Corollary
:$\forall x, y \in \R: -\paren {x - y} = -x + y$
Real Division by One
:$\forall x, y \in \R: \dfrac x 1 = x$
Reciprocal of Real Number is Non-Zero
:$\forall x \in \R: x \ne 0 \implies \dfrac 1 x \ne 0$
Differentiable Function with Bounded Derivative is Absolutely Continuous
Let $a, b$ be real numbers with $a < b$. Let $f: \closedint a b \to \R$ be a continuous function. Let $f$ be differentiable on $\openint a b$, with bounded derivative. Then $f$ is absolutely continuous.
Limit Points in Open Extension Space/Subset
Let $U \subseteq S^*_p$. Then $p$ is a limit point of $U$.
Polynomial Forms is PID Implies Coefficient Ring is Field
Let $D$ be an integral domain. Let $D \sqbrk X$ be the ring of polynomial forms in $X$ over $D$. Let $D sqbrk X$ be a principal ideal domain; Then $D$ is a field.
Sum of Strictly Positive Real Numbers is Strictly Positive
:$x, y \in \R_{>0} \implies x + y \in \R_{>0}$
Lagrange's Theorem (Number Theory)
Let $f$ be a polynomial in one variable of degree $n$ over $\Z_p$ for some prime $p$. Then $f$ has at most $n$ roots in $\Z_p$.
Finite Intersection of Regular Open Sets is Regular Open
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 open in $T$. That is: :$\forall i \in \set {1, 2, \dotsc, n}: H_i = H_i^{- \circ}$ where $H_i^{- \circ}$ denotes the interior of the closure of $H_i$. Then $\displaystyle \bigcap_{i \mathop = 1}^n H_i$ is regular open in $T$. That is: :$\displaystyle \bigcap_{i \mathop = 1}^n H_i = \paren {\bigcap_{i \mathop = 1}^n H_i}^{- \circ}$
Closure of Subspace of Normed Vector Space is Subspace
Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $Y \subseteq X$ be a subspace of $X$. Let $Y^-$ be the closure of $Y$. Then $Y^- \subseteq X$ is also a subspace of $X$.
Not every Open Set is F-Sigma Set
Let $T = \struct {S, \tau}$ be a topological space. Let $V$ be an open set of $T$. Then it is not necessarily the case that $V$ is an $F_\sigma$ set of $T$.
Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space
Let $\struct {X, \norm {\, \cdot \,}}$ is a normed vector space. Let $D \subseteq X$ be a subset of $X$. Let $D^-$ be the closure of $D$. Then $D$ is dense iff $D^- = X$.
Tychonoff Space is Preserved under Homeomorphism
If $T_A$ is a Tychonoff (completely regular) space, then so is $T_B$.
Normal Space is Preserved under Homeomorphism
If $T_A$ is a normal space, then so is $T_B$.
Left and Right Inverses of Square Matrix over Field are Equal
Let $\Bbb F$ be a field, usually one of the standard number fields $\Q$, $\R$ or $\C$. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\map \MM n$ denote the matrix space of order $n$ square matrices over $\Bbb F$. Let $\mathbf B$ be a left inverse matrix of $\mathbf A$. Then $\mathbf B$ is also a right inverse matrix of $\mathbf A$. Similarly, let $\mathbf B$ be a right inverse matrix of $\mathbf A$. Then $\mathbf B$ is also a right inverse matrix of $\mathbf A$.
Principle of Finite Induction/Peano Structure
Let $\struct {P, s, 0}$ be a Peano structure. Let $S \subseteq P$. Suppose that: :$(1): \quad 0 \in S$ :$(2): \quad \forall n: n \in S \implies \map s n \in S$ Then: :$S = P$
System of Simultaneous Equations may have Multiple Solutions
Let $S$ be a system of simultaneous equations. Then it is possible that $S$ may have a solution set which is a singleton.
Fibonacci Number as Sum of Binomial Coefficients
{{begin-eqn}} {{eqn | lo= \forall n \in \Z_{>0}:       | l = F_n       | r = \sum_{k \mathop = 0}^{\floor {\frac {n - 1} 2} } \dbinom {n - k - 1} k       | c =  }} {{eqn | r = \binom {n - 1} 0 + \binom {n - 2} 1 + \binom {n - 3} 2 + \dotsb + \binom {n - j} {j - 1} + \binom {n - j - 1} j       | c = where $j = \floor {\frac {n - 1} 2}$ }} {{end-eqn}}
Countably Metacompact Lindelöf Space is Metacompact
Let $T = \struct {S, \tau}$ be a Lindelöf space which is also countably metacompact. Then $T$ is metacompact.
Dirichlet's Approximation Theorem
Let $\alpha, x \in \R$. Then there exist integers $a, q$ such that: : $\gcd \left\{{a, q}\right\} = 1$, $1 \le q \le x$ and: :$\left|{\alpha - \dfrac a q}\right| \le \dfrac 1 {q x}$
Secant in terms of Tangent
{{begin-eqn}} {{eqn | l = \sec x       | r = +\sqrt {\tan ^2 x + 1}       | c = if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ }} {{eqn | l = \sec x       | r = -\sqrt {\tan ^2 x + 1}       | c = if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ }} {{end-eqn}}
Compact Hausdorff Topology is Minimal Hausdorff
Let $T = \struct {S, \tau}$ be a Hausdorff space which is compact. Then $\tau$ is the minimal subset of the power set $\powerset S$ such that $T$ is a Hausdorff space.
Compact Hausdorff Topology is Maximally Compact
Let $T = \struct {S, \tau}$ be a Hausdorff space which is compact. Then $\tau$ is maximally compact.
Cosine is Reciprocal of Secant
:$\cos \theta = \dfrac 1 {\sec \theta}$
Column Equivalence is Equivalence Relation
Column equivalence is an equivalence relation.
Cosine in terms of Sine
{{begin-eqn}} {{eqn | l = \cos x       | r = +\sqrt {1 - \sin^2 x}       | c = if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ }} {{eqn | l = \cos x       | r = -\sqrt {1 - \sin^2 x}       | c = if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ }} {{end-eqn}}
Sign of Cosecant
Let $x$ be a real number. Then: {{begin-eqn}} {{eqn | l = \csc x       | o = >       | r = 0       | c = if there exists an integer $n$ such that $2 n \pi < x < \paren {2 n + 1} \pi$ }} {{eqn | l = \csc x       | o = <       | r = 0       | c = if there exists an integer $n$ such that $\paren {2 n + 1} \pi < x < \paren {2 n + 2} \pi$ }} {{end-eqn}} where $\csc$ is the real cosecant function.
Square Root of Number Minus Square Root/Proof 2
Let $a$ and $b$ be (strictly) positive real numbers such that $a^2 - b > 0$. Then: {{:Square Root of Number Minus Square Root}}
Atlas Belongs to Unique Differentiable Structure
Let $M$ be a locally Euclidean space of dimension $d$. Let $\mathcal A$ be an atlas on $M$. Then there exists a unique differentiable structure $\mathcal F$ on $M$ with $\mathcal A \in \mathcal F$.
Topological Space with One Quasicomponent is Connected
Let $T = \struct {S, \tau}$ be a topological space which has one quasicomponent. Then $T$ is connected.
Multiple of Column Added to Column of Determinant
Let $\mathbf A = \begin {bmatrix} a_{1 1} & \cdots & a_{1 r} & \cdots & a_{1 s} & \cdots & a_{1 n} \\ a_{2 1} & \cdots & a_{2 r} & \cdots & a_{2 s} & \cdots & a_{2 n} \\  \vdots & \ddots &  \vdots & \ddots &  \vdots & \ddots &  \vdots \\ a_{n 1} & \cdots & a_{n r} & \cdots & a_{n s} & \cdots & a_{n n} \\ \end {bmatrix}$ be a square matrix of order $n$. Let $\map \det {\mathbf A}$ denote the determinant of $\mathbf A$. Let $\mathbf B = \begin{bmatrix} a_{1 1} & \cdots & a_{1 r} + \lambda a_{1 s} & \cdots & a_{1 s} & \cdots & a_{1 n} \\ a_{2 1} & \cdots & a_{2 r} + \lambda a_{2 s} & \cdots & a_{2 s} & \cdots & a_{2 n} \\  \vdots & \ddots &                    \vdots & \ddots &  \vdots & \ddots &  \vdots \\ a_{n 1} & \cdots & a_{n r} + \lambda a_{n s} & \cdots & a_{n s} & \cdots & a_{n n} \\ \end{bmatrix}$. Then $\map \det {\mathbf B} = \map \det {\mathbf A}$. That is, the value of a determinant remains unchanged if a constant multiple of any column is added to any other column.
Integral Closure is Integrally Closed
Let $A \subseteq B$ be an extension of commutative rings with unity. Let $C$ be the integral closure of $A$ in $B$. Then $C$ is integrally closed.
Determinant of Lower Triangular Matrix
Let $\mathbf T_n$ be a lower triangular matrix of order $n$. Let $\map \det {\mathbf T_n}$ be the determinant of $\mathbf T_n$. Then $\map \det {\mathbf T_n}$ is equal to the product of all the diagonal elements of $\mathbf T_n$. That is: :$\displaystyle \map \det {\mathbf T_n} = \prod_{k \mathop = 1}^n a_{k k}$
Relationship between Component Types
Let $T = \struct {S, \tau}$ be a topological space. Let $p \in S$. Let: : $A$ be the arc component of $p$ : $P$ be the path component of $p$ : $C$ be the component of $p$ : $Q$ be the quasicomponent of $p$. Then: :$A \subseteq P \subseteq C \subseteq Q$ In general, the inclusions do not hold in the other direction.
Ultraconnected Space is T4
Let $T = \struct {S, \tau}$ be a topological space which is ultraconnected. Then $T$ is a $T_4$ space.
Squares Ending in n Occurrences of m-Digit Pattern
Suppose there exists some integer $x$ such that $x^2$ ends in some $m$-digit pattern ending in an odd number not equal to $5$ and is preceded by another odd number, i.e.: :$\exists x \in \Z: x^2 \equiv \sqbrk {1 a_1 a_2 \cdots a_m} \pmod {2 \times 10^m}$ where $a_m$ is odd, $a_m \ne 5$ and $m \ge 1$. Then for any $n \ge 1$, there exists some integer with not more than $m n$-digits such that its square ends in $n$ occurrences of the $m$-digit pattern.
Locally Arc-Connected Space is Locally Path-Connected
Let $T = \struct {S, \tau}$ be a topological space which is locally arc-connected. Then $T$ is also locally path-connected.
Unique Factorization Domain is Integrally Closed
Let $A$ be a unique factorization domain (UFD). Then $A$ is integrally closed.
Lindelöf's Lemma
Let $C$ be a set of open real sets. Let $S$ be a real set that is covered by $C$. Then there exists a countable subset of $C$ that covers $S$.
Mellin Transform of Heaviside Step Function
Let $c$ be a constant real number. Let $u_c \left({t}\right)$ be the Heaviside step function. Let $\mathcal M$ be the Mellin transform. Then: :$\mathcal M \left\{ {u_c \left({t}\right)}\right\} \left({s}\right) = - \dfrac {c^s} s$ for $c > 0, \Re \left({s}\right) < 0$.
Complex Numbers as External Direct Product
Let $\struct {\C_{\ne 0}, \times}$ be the group of non-zero complex numbers under multiplication. Let $\struct {\R_{> 0}, \times}$ be the group of positive real numbers under multiplication. Let $\struct {K, \times}$ be the circle group. Then: :$\struct {\C_{\ne 0}, \times} \cong \struct {\R_{> 0}, \times} \times \struct {K, \times}$ {{explain|It is apparent that the second $\times$ is Cartesian product, but this is not obvious.}}
Definite Integral from 0 to Half Pi of Square of Logarithm of Cosine x
:$\displaystyle \int_0^{\pi/2} \paren {\map \ln {\cos x} }^2 \rd x = \frac \pi 2 \paren {\ln 2}^2 + \frac {\pi^3} {24}$
Metric Space Completeness is Preserved by Isometry
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Let $\phi: M_1 \to M_2$ be an isometry. If $M_1$ is complete then so is $M_2$.
3 Proper Integer Heronian Triangles whose Area and Perimeter are Equal
There are exactly $3$ proper integer Heronian triangles whose area and perimeter are equal. These are the triangles whose sides are: :$\tuple {6, 25, 29}$ :$\tuple {7, 15, 20}$ :$\tuple {9, 10, 17}$
Multiplication is Superfunction
The function $f: \C \to \C$, defined as: :$\map f z = z \times c$ is a superfunction for any complex number $c$.
Integration by Substitution/Definite Integral
The definite integral of $f$ from $a$ to $b$ can be evaluated by: :$\displaystyle \int_{\map \phi a}^{\map \phi b} \map f t \rd t = \int_a^b \map f {\map \phi u} \dfrac \d {\d u} \map \phi u \rd u$ where $x = \map \phi u$.
Discrete Space is Non-Meager
Let $T = \left({S, \tau}\right)$ be a discrete topological space. Then $T$ is non-meager.
Subset of Indiscrete Space is Dense-in-itself
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Let $H \subseteq S$ be a subset of $S$ containing more than one point. Then $H$ is dense-in-itself.
Indiscrete Space is Non-Meager
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Then $T$ is non-meager.
Equivalence of Definitions of Minimal Element
Let $\struct {S, \preceq}$ be an ordered set. Let $T \subseteq S$ be a subset of $S$. {{TFAE|def = Minimal Element}}
Non-Zero Natural Numbers under Addition form Semigroup
Let $\N_{>0}$ be the set of natural numbers without zero, that is: :$\N_{>0} = \N \setminus \set 0$ Let $+$ denote natural number addition. The structure $\struct {\N_{>0}, +}$ forms a semigroup.
Indiscrete Space is Second-Countable
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Then  $T$ is a second-countable space.
Doubling the Cube by Compass and Straightedge Construction is Impossible
There is no compass and straightedge construction to allow a cube to be constructed whose volume is double that of a given cube.
Derivative of Inverse Hyperbolic Cotangent Function
:$\map {\dfrac \d {\d x} } {\coth^{-1} u} = \dfrac {-1} {u^2 - 1} \dfrac {\d u} {\d x}$  where $\size u > 1$
Indiscrete Space is T4
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Then $T$ is a $T_4$ space.
Derivative of Cosine of Function
:$\map {\dfrac \d {\d x} } {\cos u} = -\sin u \dfrac {\d u} {\d x}$
Equation of Circle/Polar
: $r^2 - 2 r r_0 \map \cos {\theta - \varphi} + \paren {r_0}^2 = R^2$
Singleton Partition yields Indiscrete Topology
Let $S$ be a set which is not empty. Let $\PP$ be the (trivial) singleton partition $\set S$ on $S$. Then the partition topology on $\PP$ is the indiscrete topology.
Derivative of Even Function is Odd
Let $f$ be a differentiable real function such that $f$ is even. Then its derivative $f'$ is an odd function.
Graph of Quadratic describes Parabola/Corollary 2
The locus of the equation of the square root function on the non-negative reals: :$\forall x \in \R_{\ge 0}: \map f x = \sqrt x$ describes half of a parabola.
Westwood's Puzzle
:500px Take any rectangle $ABCD$ and draw the diagonal $AC$. Inscribe a circle $GFJ$ in one of the resulting triangles $\triangle ABC$. Drop perpendiculars $IEF$ and $HEJ$ from the center of this incircle $E$ to the sides of the rectangle. Then the area of the rectangle $DHEI$ equals half the area of the rectangle $ABCD$.
Number of Parameters of ARMA Model
Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$ Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$: :$\tilde z_t = z_t - \mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$ Let $M$ be an '''ARMA model''' on $S$ of order $p$: :$\tilde z_t = \phi_1 \tilde z_{t - 1} + \phi_2 \tilde z_{t - 2} + \dotsb + \phi_p \tilde z_{t - p} + a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$ Then $M$ has $p + q + 2$ parameters.
Summation Formula for Polygonal Numbers
Let $P \left({k, n}\right)$ be the $n$th $k$-gonal number. Then: : $\displaystyle P \left({k, n}\right) = \sum_{j \mathop = 1}^n \left({\left({k - 2}\right) \left({j - 1}\right) + 1}\right)$
ARIMA Model subsumes Moving Average Model
Let $S$ be a stochastic process based on an equispaced time series. Let $M$ be a moving average model for $S$. Then $M$ is also an implementation of an ARIMA model.
Infinite Particular Point Space is not Strongly Locally Compact
Let $T = \struct {S, \tau_p}$ be an infinite particular point space. Then $T$ is not strongly locally compact.
Kronecker’s Theorem
Let $K$ be a field. Let $f$ be a polynomial over $K$ of degree $n \ge 1$. Then there exists a finite extension $L / K$ of $K$ such that $f$ has at least one root in $L$. Moreover, we can choose $L$ such that the degree $\index L K$ of $L / K$ satisfies $\index L K \le n$. {{explain|Work out exactly which definitions of Polynomial and Degree are appropriate here.}}
Cancellation Law for Field Product
Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $a, b, c \in F$. Then: :$a \times b = a \times c \implies a = 0_F \text { or } b = c$
Edge is Bridge iff in All Spanning Trees
Let $G$ be a simple graph. Let $e$ be an edge of $G$. Then $e$ is a bridge in $G$  $e$ belongs to every spanning tree for $G$.
Leigh.Samphier/Sandbox/Set Difference of Distinct Equal Cardinality Sets is Not Empty
Let $S$ and $T$ be distinct finite sets. Let $\card S = \card T$. Then: :$S \setminus T \ne \O$
Particular Point Space is Locally Path-Connected
Let $T = \struct {S, \tau_p}$ be a particular point space. Then $T$ is locally path-connected.
Derivative of Inverse Hyperbolic Sine of x over a/Corollary 2
:$\map {\dfrac \d {\d x} } {\ln \size {x - \sqrt {x^2 + a^2} } } = -\dfrac 1 {\sqrt {x^2 + a^2} }$