hcju/proofwikips huggingface dataset:
Strictly Positive Rational Numbers under Multiplication form Subgroup of Non-Zero Rational Numbers
Let $\Q_{> 0}$ be the set of strictly positive rational numbers, that is $\Q_{> 0} = \set { x \in \Q: x > 0}$. The structure $\struct {\Q_{> 0}, \times}$ is a subgroup of $\struct {\Q_{\ne 0}, \times}$, where $\Q_{\ne 0}$ is the set of rational numbers without zero: $\Q_{\ne 0} = \Q \setminus \set 0$.
Union of Upper Sets is Upper
Let $\left({S, \preceq}\right)$ be a preordered set. Let $A$ be a set of subsets of $S$ such that :$\forall X \in A: X$ is an upper set. Then: :$\bigcup A$ is also an upper set.
Set of Rotations in Space about Fixed Point forms Infinite Group
Let $\SS$ be a rigid body in space. Let $O$ be a fixed point in space. The set of all rotations of $\SS$ through some line through $O$ forms an infinite group.
Group of Rotations about Fixed Point is not Abelian
Let $\SS$ be a rigid body in space. Let $O$ be a fixed point in space. Let $\GG$ be the group of all rotations of $\SS$ around $O$. Then $\GG$ is not an abelian group.
Union of Filtered Sets is Filtered
Let $\left({S, \preceq}\right)$ be a preordered set. Let $A$ be a set of subsets of $S$ such that :$\forall X \in A: X$ is filtered and :$\forall X, Y \in A: \exists Z \in A: X \cup Y \subseteq Z$ Then: :$\bigcup A$ is also filtered.
Special Orthogonal Group is Group
Let $k$ be a field. The $n$th orthogonal group on $k$ is a group.
Unit Matrix is its own Inverse
The inverse of the unit matrix $\mathbf I_n$ of order $n$ is $\mathbf I_n$. That is, a unit matrix it its own inverse.
Way Above Closure is Subset of Upper Closure of Element
Let $\left({S, \preceq}\right)$ be an ordered set. Let $x \in S$. Then $x^\gg \subseteq x^\succeq$ where :$x^\gg$ denotes the way above closure of $x$, :$x^\succeq$ denotes the upper closure of $x$.
Diagonal Matrix is Symmetric
Let $D$ be a diagonal matrix. Then $D$ is symmetric.
Upper Way Below Open Subset Complement is Non Empty implies There Exists Maximal Element of Complement
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice. Let $X$ be upper way below open subset of $S$. Let $x \in S$ such that :$x \in \relcomp S X$ Then :$\exists m \in S: x \preceq m \land m = \max \relcomp S X$
Unit Matrix is Orthogonal
The unit matrix $\mathbf I_n$ of order $n$ is orthogonal.
Unit Matrix is Proper Orthogonal
The unit matrix $\mathbf I_n$ of order $n$ is proper orthogonal.
Inverse of Orthogonal Matrix is Orthogonal
Let $\mathbf A$ be an orthogonal matrix. Then its inverse $\mathbf A^{-1}$ is also orthogonal.
Inverse of Inverse of Matrix
Let $\mathbf A$ be an invertible matrix. Then: :$\paren {\mathbf A^{-1} }^{-1} = \mathbf A$ That is, an invertible matrix equals the inverse of its inverse.
Inverse of Proper Orthogonal Matrix is Proper Orthogonal
Let $\mathbf A$ be a proper orthogonal matrix. Then its inverse $\mathbf A^{-1}$ is also proper orthogonal.
Special Orthogonal Group is Subgroup of Orthogonal Group
Let $k$ be a field. Let $\map {\operatorname O} {n, k}$ be the $n$th orthogonal group on $k$. Let $\map {\operatorname {SO} } {n, k}$ be the $n$th special orthogonal group on $k$. Then $\map {\operatorname {SO} } {n, k}$ is a subgroup of $\map {\operatorname O} {n, k}$.
Negative Matrix is Inverse for Hadamard Product
Let $\struct {G, \cdot}$ be a group whose identity is $e$. Let $\map {\MM_G} {m, n}$ be a $m \times n$ matrix space over $\struct {G, \cdot}$. Let $\mathbf A$ be an element of $\map {\MM_G} {m, n}$. Let $-\mathbf A$ be the negative of $\mathbf A$. Then $-\mathbf A$ is the inverse for the operation $\circ$, where $\circ$ is the Hadamard product.
Left Regular Representation wrt Left Cancellable Element on Finite Semigroup is Bijection
Let $\struct {S, \circ}$ be a finite semigroup. Let $a \in S$ be left cancellable. Then the left regular representation $\lambda_a$ of $\struct {S, \circ}$ with respect to $a$ is a bijection.
Singleton is Chain
Let $\left({S, \preceq}\right)$ be an ordered set. Let $x \in S$. Then $\left\{ {x}\right\}$ is a chain of $\left({S, \preceq}\right)$.
Identity Elements occupy Diagonal of Cayley Table in Inverse Row Form
Let $\struct {G, \circ}$ be a finite group. Let $\CC$ be a Cayley table for $\struct {G, \circ}$ presented in inverse row form. Then all the entries in the main diagonal of $\CC$ are instances of the identity element.
Scaling preserves Modulo Addition
Let $m \in \Z_{> 0}$. Let $x, y, c \in \Z$. Let $x \equiv y \pmod m$. Then: :$c x \equiv c y \pmod m$
Modulo Addition is Linear
Let $m \in \Z_{> 0}$. Let $x_1, x_2, y_1, y_2, c_1, c_2 \in \Z$. Let: :$x_1 \equiv y_1 \pmod m$ :$x_2 \equiv y_2 \pmod m$ Then: :$c_1 x_1 + c_2 x_2 \equiv c_1 y_1 + c_2 y_2 \pmod m$
Euler Phi Function of Prime
Let $p$ be a prime number $p > 1$. Then: :$\map \phi p = p - 1$ where $\phi: \Z_{>0} \to \Z_{>0}$ is the Euler $\phi$ function.
Chain is Directed
Let $\struct {S, \preceq}$ be an ordered set. Let $C$ be a non-empty chain of $S$. Then $C$ is directed.
Reduced Residue System Modulo Prime
Let $p$ be a prime number. The reduced residue system modulo $p$ contains $p - 1$ elements: :$Z'_p = \set {\eqclass 1 m, \eqclass 2 m, \ldots, \eqclass {p - 1} m}$ and so can be defined as: :$Z'_p = Z_p \setminus \set {\eqclass 0 m}$
Modulo Multiplication on Reduced Residue System is Closed
Let $m \in \Z_{> 0}$ be a (strictly) positive integer. Let $\Z'_m$ be the reduced residue system modulo $m$: :$\Z'_m = \set {\eqclass k m \in \Z_m: k \perp m}$ Let $S = \struct {\Z'_m, \times_m}$ be the algebraic structure consisting of $\Z'_m$ under the modulo multiplication. Then $S$ is closed, in the sense that: :$\forall a, b \in \Z'_m: a \times_m b \in \Z'_m$
Modulo Multiplication on Reduced Residue System is Cancellable
Let $m \in \Z_{> 0}$ be a (strictly) positive integer. Let $\Z'_m$ be the reduced residue system modulo $m$: :$\Z'_m = \set {\eqclass k m \in \Z_m: k \perp m}$ Let $S = \struct {\Z'_m, \times_m}$ be the algebraic structure consisting of $\Z'_m$ under modulo multiplication. Then $\times_m$ is cancellable, in the sense that: :$\forall a, b, c \in \Z'_m: a \times_m c = b \times_m c \implies a = b$ and: :$\forall a, b, c \in \Z'_m: c \times_m a = c \times_m b \implies a = b$
Order Generating iff Every Element is Infimum
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice. Let $X$ be a subset of $S$. Then :$X$ is order generating  :$\forall x \in S: \exists Y \subseteq X: x = \inf Y$
Weierstrass Factorization Theorem
Let $f$ be an entire function. Let $0$ be a zero of $f$ of multiplicity $m \ge 0$. Let the sequence $\sequence {a_n}$ consist of the nonzero zeroes of $f$, repeated according to multiplicity.
Equivalence of Axiom Schemata for Finite Group
The following axiom schemata for the definition of a finite group are logically equivalent:
Order Generating iff Every Superset Closed on Infima is Whole Space
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $X$ be a subset of $S$. Then :$X$ is order generating  :$\forall Y \subseteq S: Y \supseteq X \land \left({\forall Z \subseteq Y: \inf Z \in Y}\right) \implies Y = S$
Group Isomorphism Preserves Inverses
Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group isomorphism. Let: :$e_G$ be the identity of $\struct {G, \circ}$ :$e_H$ be the identity of $\struct {H, *}$. Then: : $\forall g \in G: \map \phi {g^{-1} } = \paren {\map \phi g}^{-1}$
Preimage of Composite Relation
Let $\RR_1 \subseteq S_1 \times T_1$ and $\RR_2 \subseteq S_2 \times T_2$ be relations. Let $\RR_2 \circ \RR_1 \subseteq S_1 \times T_2$ be the composition of $\RR_1$ and $\RR_2$. Then the preimage of $\RR_2 \circ \RR_1$ is given by: :$\Preimg {\RR_2 \circ \RR_1} = \Preimg {\Img {\RR_1} \cap \Preimg {\RR_2} }$
Image of Composite Relation
Let $\RR_1 \subseteq S_1 \times T_1$ and $\RR_2 \subseteq S_2 \times T_2$ be relations. Let $\RR_2 \circ \RR_1 \subseteq S_1 \times T_2$ be the composition of $\RR_1$ and $\RR_2$. Then the image of $\RR_2 \circ \RR_1$ is given by: :$\Img {\RR_2 \circ \RR_1} = \Img {\Img {\RR_1} \cap \Preimg {\RR_2} }$
Weierstrass Product Theorem
Let $\sequence {a_k}$ be a sequence of non-zero complex numbers such that: :$\cmod {a_n} \to \infty$ as $n \to \infty$ Let $\sequence {p_n}$ be a sequence of non-negative integers for which the series: :$\displaystyle \sum_{n \mathop = 1}^\infty \size {\dfrac r {a_n} }^{1 + p_n}$ converges for every $r \in \R_{> 0}$. Let: :$\displaystyle \map f z = \prod_{n \mathop = 1}^\infty \map {E_{p_n} } {\frac z {a_n} }$ where $E_{p_n}$ are Weierstrass elementary factors. Then $f$ is entire and its zeroes are the points $a_n$, counted with multiplicity.
Isomorphism between Gaussian Integer Units and Rotation Matrices Order 4
Let $\struct {U_\C, \times}$ be the group of Gaussian integer units under complex multiplication. Let $\struct {R_4, \times}$ be the group of rotation matrices of order $4$ under modulo addition. Then $\struct {U_\C, \times}$ and $\struct {R_4, \times}$ are isomorphic algebraic structures.
Isomorphism between Gaussian Integer Units and Reduced Residue System Modulo 5 under Multiplication
Let $\struct {U_\C, \times}$ be the group of Gaussian integer units under complex multiplication. Let $\struct {\Z'_5, \times_5}$ be the multiplicative group of reduced residues modulo $5$. Then $\struct {U_\C, \times}$ and $\struct {\Z'_5, \times_5}$ are isomorphic algebraic structures.
Group Generated by Reciprocal of z and 1 minus z
Let $\struct {S, \circ}$ denote the '''group generated by $\dfrac 1 z$ and $1 - z$'''. Then $\struct {S, \circ}$ is a finite group of order $6$.
Group Generated by Reciprocal of z and Minus z is Klein Four-Group
Let $K_4$ denote the Klein $4$-group. Let $S$ be the group generated by $1 / z$ and $-z$. Then $K_4$ and $S$ are isomorphic algebraic structures.
Group of Reflection Matrices Order 4 is Klein Four-Group
Let $K_4$ denote the Klein $4$-group. Let $R_4$ be the Group of Reflection Matrices Order $4$. Then $K_4$ and $R_4$ are isomorphic algebraic structures.
Klein Four-Group and Group of Cyclic Group of Order 4 are not Isomorphic
The Klein $4$-group $K_4$ and the cyclic group of order $4$ $C_4$ are not isomorphic.
Order Generating iff Not Preceding implies There Exists Element Preceding and Not Preceding
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $X$ be a subset of $S$. Then :$X$ is order generating  :$\forall x, y \in S: \left({ y \npreceq x \implies \exists p \in X: x \preceq p \land y \npreceq p}\right)$
Permutation Induces Equivalence Relation/Corollary
:$i \mathrel {\mathcal R_\pi} j$  $i$ and $j$ are in the same cycle of $\pi$.
Order of Group Element not less than Order of Power
Let $\left({G, \circ}\right)$ be a group whose identity is $e$. Let $g \in G$ be an element of $g$. Let $\left\lvert{g}\right\rvert$ denote the order of $g$ in $G$. Then: :$\forall m \in \Z: \left\lvert{g^m}\right\rvert \le \left\lvert{g}\right\rvert$ where $g^m$ denotes the $m$th power of $g$ in $G$.
Order of Group Element equals Order of Coprime Power
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $g \in G$ be an element of $g$. Let $\order g$ denote the order of $g$ in $G$. Then: :$\forall m \in \Z: \order {g^m} = \order g \iff m \perp \order g$ where: :$g^m$ denotes the $m$th power of $g$ in $G$ :$\perp$ denotes coprimality.
Unique Composition of Group Element whose Order is Product of Coprime Integers
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $g \in G$ be an element of $g$. Let: :$\order g = m n$ where: :$\order g$ denotes the order of $g$ in $G$ :$m$ and $n$ are coprime integers. Then $g$ can be expressed uniquely as the product of two commuting elements $a$ and $b$ of order $m$ and $n$ respectively.
Order of Cyclic Group equals Order of Generator
Let $G$ be a finite cyclic group which is generated by $a \in G$. Then: :$\order a = \order G$ where: :$\order a$ denotes the order of $a$ in $G$ :$\order G$ denotes the order of $G$.
Group whose Order equals Order of Element is Cyclic
Let $G$ be a finite group of order $n$. Let $g \in G$ have order $n$. Then $G$ is a cyclic group which is generated by $g$.
Integer Powers of 2 under Multiplication form Infinite Abelian Group
Let $S$ be the set of integers defined as: :$S = \set {2^k: k \in \Z}$ Then $\struct {S, \times}$ is an infinite abelian group.
Numbers of form 1 + 2m over 1 + 2n form Infinite Abelian Group under Multiplication
Let $S$ be the set of integers defined as: :$S = \set {\dfrac {1 + 2 m} {1 + 2 n}: m, n \in \Z}$ Then $\struct {S, \times}$ is an infinite abelian group.
Superset of Order Generating is Order Generating
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $X, Y$ be subsets of $S$ such that :$X$ is order generating and :$X \subseteq Y$ Then $Y$ is order generating.
Modulus 1 Rational Argument Complex Numbers under Multiplication form Infinite Abelian Group
Let $S$ be the set defined as: :$S = \set {\cos \theta + i \sin \theta: \theta \in \Q}$ Then the algebraic structure $\struct {S, \times}$ is an infinite abelian group.
Inversion Mapping is Automorphism iff Group is Abelian
Let $\struct {G, \circ}$ be a group. Let $\iota: G \to G$ be the inversion mapping on $G$, defined as: :$\forall g \in G: \map \iota g = g^{-1}$ Then $\iota$ is an automorphism  $G$ is abelian.
Even Order Group has Odd Number of Order 2 Elements
Let $G$ be a group whose identity is $e$. Let $G$ be of even order. Then $G$ has an odd number of elements of order $2$.
Roots of Resolvent of Cubic
Let $P$ be the cubic equation: : $a x^3 + b x^2 + c x + d = 0$ with $a \ne 0$ Let $\map r P$ be the resolvent equation of $P$, given by: : $u^6 - 2 R u^3 - Q^3$ Let the roots of $P$ be $\alpha_1, \alpha_2, \alpha_3$. Then the roots of $\map r P$ can be expressed as: {{begin-eqn}} {{eqn | l = v       | r = \frac 1 3 \paren {\alpha_1 + \omega \alpha_2 + \omega^2 \alpha_3} }} {{eqn | l = \omega v       | r = \frac 1 3 \paren {\alpha_3 + \omega \alpha_1 + \omega^2 \alpha_2}       | c =  }} {{eqn | l = \omega^2 v       | r = \frac 1 3 \paren {\alpha_2 + \omega \alpha_3 + \omega^2 \alpha_1}       | c =  }} {{eqn | l = u       | r = \frac 1 3 \paren {\alpha_1 + \omega \alpha_3 + \omega^2 \alpha_2} }} {{eqn | l = \omega u       | r = \frac 1 3 \paren {\alpha_2 + \omega \alpha_1 + \omega^2 \alpha_3}       | c =  }} {{eqn | l = \omega^2 u       | r = \frac 1 3 \paren {\alpha_3 + \omega \alpha_2 + \omega^2 \alpha_1}       | c =  }} {{end-eqn}} where $\omega = -\dfrac {-1 + \sqrt {-3} } 2$ is one of the primitive (complex) cube roots of $1$.
Roots of Complex Number/Examples/Cube Roots
Let $z := \polar {r, \theta}$ be a complex number expressed in polar form, such that $z \ne 0$. Then the complex cube roots of $z$ are given by: :$z^{1 / 3} = \set {r^{1 / 3} \paren {\map \cos {\dfrac {\theta + 2 \pi k} 3} + i \, \map \sin {\dfrac {\theta + 2 \pi k} 3} }: k \in \set {0, 1, 2} }$ There are $3$ distinct such complex cube roots. These can also be expressed as: :$z^{1 / 3} = \set {r^{1 / 3} e^{i \paren {\theta + 2 \pi k} / 3}: k \in \set {0, 1, 2} }$ or: :$z^{1 / 3} = \set {r^{1 / 3} e^{i \theta / 3} \omega^k: k \in \set {0, 1, 2} }$ where $\omega = e^{2 i \pi / 3} = -\dfrac 1 2 + \dfrac {i \sqrt 3} 2$ is the first cube root of unity.
Roots of Complex Number/Corollary
Let $z := \polar {r, \theta}$ be a complex number expressed in polar form, such that $z \ne 0$. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $w$ be one of the complex $n$th roots of $z$. Then the $n$th roots of $z$ are given by: :$z^{1 / n} = \set {w \epsilon^k: k \in \set {1, 2, \ldots, n - 1} }$ where $\epsilon$ is a primitive $n$th root of unity.
Top is Prime Element
Let $L = \left({S, \wedge, \preceq}\right)$ be a bounded above meet semilattice. Then $\top$ is a prime element where $\top$ denotes the greatest element of $L$.
Characterization of Prime Element in Meet Semilattice
Let $L = \struct {S, \wedge, \preceq}$ be a meet semilattice. Let $p \in S$, Then: :$p$ is prime element : :for all non-empty finite subsets $A$ of $S$: ::if $\inf A \preceq p$, then there exists element $x$ of $A$ such that $x \preceq p$.
Prime Element is Meet Irreducible
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a lattice. Let $p \in S$. Let $p$ be a prime element of $L$. Then $p$ is meet irreducible in $L$.
Prime Element iff Meet Irreducible in Distributive Lattice
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a distributive lattice. Let $p \in S$. Then $p$ is a prime element  $p$ is meet irreducible.
Multiple Equilibrium Points all have Equal Payoffs
Let $G$ be a two-person zero-sum game. Let $G$ have more than one equilibrium point. Then every equilibrium point of $G$ has the same payoff.
Definition:Information of Game
The '''information''' of a game is the knowledge each player has of the history of all the moves of all the players of the game.
Multigraph with Finite Vertex Set may not be Finite
Let $G$ be a multigraph. Let the vertex set of $G$ be finite. Then it is not necessarily the case that $G$ is also finite.
Modulus Larger than Real Part
:$\cmod z \ge \size {\map \Re z}$
Modulus Larger than Imaginary Part
:$\cmod z \ge \size {\map \Im z}$
Product Space is T0 iff Factor Spaces are T0/General Result
Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $\displaystyle T = \struct{S, \tau} = \prod \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$. Then $T$ is a $T_0$ (Kolmogorov) space  each of $\struct{S_\alpha, \tau_\alpha}$ is a $T_0$ (Kolmogorov) space.
Separation Properties Preserved under Topological Product/Corollary
$T = \struct {S, \tau}$ has one of the following properties  each of $\struct {S_i, \tau_i}$ has the same property: :Regular Property :Tychonoff (Completely Regular) Property If $T = \struct {S, \tau}$ has one of the following properties then each of $\struct {S_i, \tau_i}$ has the same property: :Normal Property :Completely Normal Property but the converse does not necessarily hold.
Quotient and Remainder to Number Base/General Result
:$\left \lfloor {\dfrac n {b^s}} \right \rfloor = \left[{r_m r_{m-1} \ldots r_{s+1} r_s}\right]_b$ :$\displaystyle n \,\bmod\, {b^s} = \sum_{j \mathop = 0}^{s-1} {r_j b^j} = \left[{r_{s-1} r_{s-2} \ldots r_1 r_0}\right]_b$
Remainder is Primitive Recursive
Let $m, n \in \N$ be natural numbers. Let us define the function $\operatorname{rem}: \N^2 \to \N$: :$\map \rem {n, m} = \begin{cases} \text{the remainder when } n \text{ is divided by } m & : m \ne 0 \\ 0 & : m = 0 \end{cases}$ where the $\text{remainder}$ is as defined in the Division Theorem: :If $n = m q + r$, where $0 \le r < m$, then $r$ is the remainder. Then $\rem$ is primitive recursive.
Quotient is Primitive Recursive
Let $m, n \in \N$ be natural numbers. Let us define the function $\operatorname {quot}: \N^2 \to \N$: :$\map {\operatorname {quot} } {n, m} = \begin{cases} \text{the quotient when } n \text{ is divided by } m & : m \ne 0 \\ 0 & : m = 0 \end{cases}$ where the $\text {quotient}$ and $\text {remainder}$ are as defined in the Division Theorem: :If $n = m q + r$, where $0 \le r < m$, then $q$ is the quotient. Then $\operatorname {quot}$ is primitive recursive.
Not Every Two-Person Zero-Sum Game has Saddle Point
Not every two-person zero-sum game has a saddle point.
Two-Person Zero-Sum Game with Multiple Solutions
There exists a two-person zero-sum game with more than one solution.
Two-Person Zero-Sum Game with Finite Strategies has Solution
Let $G$ be a two-person zero-sum game. Let each player of $G$ have a finite set of strategies available. Then $G$ has at least one solution.
Matching Pennies is Completely Mixed Game
The game of matching pennies is a completely mixed game.
Value of Skew-Symmetric Game is Zero
Let $G$ be a two-person zero-sum game. Let $G$ be represented by a payoff table that is skew-symmetric. Then the value of $G$ is zero.
Prime Element iff Complement of Lower Closure is Filter
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded above lattice. Let $p \in S$ such that :$p \ne \top$ where $\top$ denotes the top of $L$. Then: :$p$ is a prime element  :$\relcomp S {p^\preceq}$ is filter in $L$ where :$p^\preceq$ denotes the lower closure of $p$ :$\relcomp S {p^\preceq}$ denotes the relative complement of $p^\preceq$ relative to $S$.
Oscillation at Point (Infimum) equals Oscillation at Point (Epsilon-Neighborhood)
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 $E_x$ be the set of $\epsilon$-neighborhoods of $x$. Let $\map {\omega_f} x$ be the oscillation of $f$ at $x$ based on $N_x$: :$\map {\omega_f} x = \inf \set {\map {\omega_f} I: I \in N_x}$ where $\map {\omega_f} I$ is the oscillation of $f$ on $I$: :$\map {\omega_f} I = \sup \set {\size {\map f y - \map f z}: y, z \in I \cap D}$ Let $\map {\omega^E_f} x$ be the oscillation of $f$ at $x$ based on $E_x$: :$\map {\omega^E_f} x = \inf \set {\map {\omega_f} I: I \in E_x}$ Then: :$\map {\omega_f} x \in \R$  $\map {\omega^E_f} x \in \R$ and, if $\map {\omega_f} x$ and $\map {\omega^E_f} x$ exist as real numbers: :$\map {\omega_f} x = \map {\omega^E_f} x$
Prime Element iff There Exists Way Below Open Filter which Complement has Maximum
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below continuous distributive lattice. Let $p \in S$ such that :$p \ne \top$ where $\top$ denotes the top of $L$. Then :$p$ is a prime element  :there exists a way below open filter $F$ in $L$: $p = \max \left({ \complement_S\left({F}\right) }\right)$
Characterization of Prime Ideal
Let $L = \struct {S, \wedge, \preceq}$ be a meet semilattice. Let $I$ be an proper ideal in $L$. Then :$I$ is a prime ideal  :$\forall x, y \in S: \paren {x \wedge y \in I \implies x \in I \lor y \in I}$
Matching Pennies is Strictly Competitive
The game of Matching Pennies is strictly competitive.
Second Price Auction has Inefficient Equilibria
A second price auction has Nash equilibria which are inefficient.
Expected-Utility Maximization Theorem
Let $P$ be a rational decision-maker. Let $A$ be a set of possible moves available to $P$. Then there exists a way of assigning values to a payoff function $u: A \to \R$ so that $P$ will always choose the move such that $u$ is a maximum.
Epic Equalizer is Isomorphism
Let $\mathbf C$ be a metacategory. Let $e: E \to C$ be the equalizer of two morphisms $f, g: C \to D$. Let $e$ be an epimorphism. Then $e$ is an isomorphism.
Infimum of Subset of Extended Real Numbers is Arbitrarily Close
Let $A \subseteq \overline \R$ be a subset of the extended real numbers. Let $b$ be an infimum (in $\R$) of $A$. Let $\epsilon \in \R_{>0}$. Then: :$\exists x \in \paren {A \cap \R}: x - b < \epsilon$
Oscillation on Set is an Extended Real Number
Let $f: D \to \R$ be a real function where $D \subseteq \R$. Let $x$ be a point in $D$. Let $I$ be a real set that contains (as an element) $x$. Let $\omega_f \left({I}\right)$ be the oscillation of $f$ on $I$: :$\omega_f \left({I}\right) = \sup \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ Then: :$\omega_f \left({I}\right) \in \overline \R_{\ge 0}$ and: :$\omega_f \left({I}\right) = \begin{cases} \text{a positive real number} & \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\} \text{is bounded above} \\ \infty & \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\} \text{is not bounded above} \end{cases}$
Oscillation on Subset
Let $f: D \to \R$ be a real function where $D \subseteq \R$. Let $x$ be a point in $D$. Let $S_x$ be a set of real sets that contain (as an element) $x$. Let $\map {\omega_f} I$ be the oscillation of $f$ on a set $I$ in $S_x$: :$\map {\omega_f} I = \sup \set {\size {\map f y - \map f z}: y, z \in I \cap D}$ Let $I \in S_x$. Let $\map {\omega_f} I \in \R$. Let $J \in S_x$ be a subset of $I$. Then: :$\map {\omega_f} J \in \R$ :$\map {\omega_f} J \le \map {\omega_f} I$
Generating Fraction for Lucas Numbers
The fraction: :$\dfrac {199} {9899}$ has a decimal expansion which contains within it the start of the Lucas sequence: :$0 \cdotp 02010 \, 30407 \, 11 \ldots$
Transcendental Numbers are Uncountable
The set of transcendental real numbers is uncountable.