hcju/proofwikips huggingface dataset:
External Direct Product of Projection with Canonical Injection/General Result
Let $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \dotsc, \struct {S_j, \circ_j}, \dotsc, \struct {S_n, \circ_n}$ be algebraic structures with identities $e_1, e_2, \dotsc, e_j, \dotsc, e_n$ respectively. Let $\displaystyle \struct {S, \circ} = \prod_{i \mathop = 1}^n \struct {S_i, \circ_i}$ be the external direct product of $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \dotsc, \struct {S_j, \circ_j}, \dotsc, \struct {S_n, \circ_n}$. Let $\pr_j: \struct {S, \circ} \to \struct {S_j, \circ_j}$ be the $j$th projection from $\struct {S, \circ}$ to $\struct {S_j, \circ_j}$. Let $\inj_j: \struct {S_j, \circ_j} \to \struct {S, \circ}$ be the canonical injection from $\struct {S_j, \circ_j}$ to $\struct {S, \circ}$. Then: :$\pr_j \circ \inj_j = I_{S_j}$ where $I_{S_j}$ is the identity mapping from $S_j$ to $S_j$.
Pre-Image Sigma-Algebra on Codomain is Sigma-Algebra
Let $X, X'$ be sets, and let $f: X \to X'$ be a mapping. Let $\Sigma$ be a $\sigma$-algebra on $X$. Denote with $\Sigma'$ the pre-image $\sigma$-algebra on the domain of $f$. Then $\Sigma'$ is a $\sigma$-algebra on $X'$.
Multiplicative Group of Reduced Residues Modulo 8 is Klein Four-Group
Let $K_4$ denote the Klein $4$-group. Let $R_4$ be the multiplicative group of reduced residues Modulo $8$. Then $K_4$ and $R_4$ are isomorphic algebraic structures.
Properties of Relation Not Preserved by Restriction
If a relation is: * serial, * non-reflexive, * non-symmetric, * non-transitive or * non-connected it is impossible to state without further information whether or not any restriction of that relation has the same properties.
Totally Ordered Set is Well-Ordered iff Subsets Contain Infima
Let $\left({S, \preccurlyeq}\right)$ be a totally ordered set. Then $\left({S, \preccurlyeq}\right)$ is a well-ordered set iff every non-empty subset of $T \subseteq S$ has an infimum such that $\inf \left({T}\right) \in T$.
Cantor Set has Zero Lebesgue Measure
Let $\mathcal C$ be the Cantor set. Let $\lambda$ be the Lebesgue measure on the Borel $\sigma$-algebra $\mathcal B \left({\R}\right)$ on $\R$. Then $\mathcal C$ is $\mathcal B \left({\R}\right)$-measurable, and $\lambda \left({\mathcal C}\right) = 0$. That is, $\mathcal C$ is a $\lambda$-null set.
Factorization Lemma/Real-Valued Function
Then a mapping $g: X \to \R$ is $\map \sigma f \, / \, \map {\mathcal B} \R$-measurable : :There exists a $\Sigma \, / \, \map {\mathcal B} \R$-measurable mapping $\tilde g: Y \to \R$ such that $g = \tilde g \circ f$ where: :$\map \sigma f$ denotes the $\sigma$-algebra generated by $f$ :$\map {\mathcal B} \R$ denotes the Borel $\sigma$-algebra on $\R$
Non-Zero-Sum Game as Zero-Sum Game
Let $G$ be a non-zero-sum game for $n$ players. Then $G$ can be modelled as a zero-sum game for $n + 1$ players.
Two-Person Zero-Sum Game is Non-Cooperative
A two-person zero-sum game necessarily has to be non-cooperative.
Simple Graph with Finite Vertex Set is Finite
Let $G$ be a simple graph. Suppose that the vertex set of $G$ is finite. Then $G$ is a finite graph. That is to say, its edge set is also finite.
Tarski's Geometry is Complete/Corollary
Tarski's geometry does not contain minimal arithmetic.
Symmetric Closure of Symmetric Relation
Let $\mathcal R$ be a relation on a set $S$. Let $\mathcal R^\leftrightarrow$ be the symmetric closure of $\mathcal R$. Then $\mathcal R = \mathcal R^\leftrightarrow$.
Eluding Game has no Saddle Point
The eluding game has no saddle point.
Prime Element iff Element Greater is Top
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a Boolean lattice. Let $p \in S$ such that :$p \ne \top$ Then :$p$ is prime element  :$\forall x \in S: \left({ p \prec x \implies x = \top }\right)$
Relation Isomorphism Preserves Antisymmetry
Let $\left({S, \mathcal R_1}\right)$ and $\left({T, \mathcal R_2}\right)$ be relational structures. Let $\left({S, \mathcal R_1}\right)$ and $\left({T, \mathcal R_2}\right)$ be (relationally) isomorphic. Then $\mathcal R_1$ is an antisymmetric relation  $\mathcal R_2$ is also an antisymmetric relation.
Anomalous Cancellation on 2-Digit Numbers
There are exactly four anomalously cancelling vulgar fractions having two-digit numerator and denominator when expressed in base $10$ notation: {{:Anomalous Cancellation on 2-Digit Numbers/Examples}}
Intersection of Strict Lower Closures in Toset
Let $\left({S, \preceq}\right)$ be a totally ordered set. Let $a,b \in S$. Then: :$a^\prec \cap b^\prec = \left({\min \left({a, b}\right)}\right)^\prec$ where: : $a^\prec$ denotes strict lower closure of $a$ : $\min$ denotes the min operation.
Intermediate Value Theorem (Topology)
Let $X$ be a connected topological space. Let $\struct {Y, \preceq, \tau}$ be a totally ordered set equipped with the order topology. Let $f: X \to Y$ be a continuous mapping. Let $a$ and $b$ are two points of $a, b \in X$ such that: :$\map f a \prec \map f b$ Let: :$r \in Y: \map f a \prec r \prec \map f b$ Then there exists a point $c$ of $X$ such that: :$\map f c = r$
Equivalence of Definitions of Upper Wythoff Sequence
The following definitions of the upper Wythoff sequence are equivalent:
Positive Integer is Sum of Consecutive Positive Integers iff not Power of 2
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then $n$ can be expressed as the sum of $2$ or more consecutive (strictly) positive integers  $n$ is not a power of $2$.
Baire-Osgood Theorem
Let $X$ be a Baire space. Let $Y$ be a metrizable topological space Let $f: X \to Y$ be a mapping which is the pointwise limit of a sequence $\left \langle{f_n}\right\rangle$ in $C \left({X, Y}\right)$. {{explain|$C \left({X, Y}\right)$}} Let $D \left({f}\right)$ be the set of points where $f$ is discontinuous. Then $D \left({f}\right)$ is a meager subset of $X$.
Ramanujan's Infinite Nested Roots
:$3 = \sqrt {1 + 2 \sqrt {1 + 3 \sqrt { 1 + \cdots} } }$
Product of Three Consecutive Integers is never Perfect Power
Let $n \in \Z_{> 1}$ be a (strictly) positive integer. Then: :$\paren {n - 1} n \paren {n + 1}$ cannot be expressed in the form $a^k$ for $a, k \in \Z$ where $k \ge 2$. That is, the product of $3$ consecutive (strictly) positive integers can never be a perfect power.
Maximal Element need not be Greatest Element
Let $\struct {S, \preccurlyeq}$ be an ordered set. Let $M \in $ be a maximal element of $S$. Then $M$ is not necessarily the greatest element of $S$.
Maximal Ideal of Division Ring
Let $\left({D, +, \circ}\right)$ be a Division Ring whose zero is $0$. :Let $\left({J, +, \circ}\right)$ be a maximal ideal of $D$. Then $J = \left\{{0}\right\}$.
Identity Mapping is Ordered Ring Automorphism
Let $\struct {S, +, \circ, \preceq}$ be an ordered ring. Then the identity mapping $I_S: S \to S$ is an ordered ring automorphism.
Number of Binary Digits in Power of 10/Example/1000
When expressed in binary notation, the number of digits in $1000$ is $10$.
Characteristic of Integral Domain is Zero or Prime
Let $\struct {D, +, \circ}$ be an integral domain. Let $\operatorname{Char} \left({D}\right)$ be the characteristic of $D$. Then $\operatorname{Char} \left({D}\right)$ is either $0$ or a prime number.
Number is Sum of Five Cubes
Let $n \in \Z$ be an integer. Then $n$ can be expressed as the sum of $5$ cubes (either positive or negative) in an infinite number of ways.
Exponential is of Exponential Order Real Part of Index
Let $\map f t = e^{\psi t}$ be the complex exponential function, where $t \in \R, \psi \in \C$. Let $a = \map \Re \psi$. Then $e^{\psi t}$ is of exponential order $a$.
Scalar Product with Multiple of Unity
:$\paren {n \cdot 1_R} \circ x = n \cdot x$ that is: :$\paren {\map {\paren {+_R}^n} {1_R} } \circ x = \map {\paren {+_G}^n} x$
Pythagorean Triangle from Fibonacci Numbers
Take $4$ consecutive Fibonacci numbers: :$F_n, F_{n + 1}, F_{n + 2}, F_{n + 3}$ Let: :$a := F_n F_{n + 3}$ :$b := 2 F_{n + 1} F_{n + 2}$ :$c := F_{2 n + 3}$ Then: :$a^2 + b^2 = c^2$ and: :$\dfrac {a b} 2 = F_n \times F_{n + 1} \times F_{n + 2} \times F_{n + 3}$ That is, if the legs of a right triangle are the product of the outer terms and twice the inner terms, then: :the hypotenuse is the Fibonacci number whose index is half the sum of the indices of the four given Fibonacci numbers. :the area is the product of the four given Fibonacci numbers.
Conditions for Homogeneity/Plane
The plane $P = \alpha_1 x_1 + \alpha_2 x_2 + \alpha_3 x_3 = \gamma$ is homogeneous iff $\gamma = 0$.
Field Homomorphism Preserves Subfields
Let $\struct {F_1, +_1, \circ_1}$ and $\struct {F_2, +_2, \circ_2}$ be fields. Let $\phi: F_1 \to F_2$ be a field homomorphism such that $\phi$ is not the trivial homomorphism. If $K$ is a subfield of $F_1$, then $\phi \sqbrk K$ is a subfield of $F_2$.
Fermat Quotient of 2 wrt p is Square iff p is 3 or 7
Let $p$ be a prime number. The Fermat quotient of $2$ with respect to $p$: :$\map {q_p} 2 = \dfrac {2^{p - 1} - 1} p$ is a square  $p = 3$ or $p = 7$.
Integer Addition is Well-Defined
Let $\struct {\N, +}$ be the semigroup of natural numbers under addition. Let $\struct {\N \times \N, \oplus}$ be the (external) direct product of $\struct {\N, +}$ with itself, where $\oplus$ is the operation on $\N \times \N$ induced by $+$ on $\N$. Let $\boxtimes$ be the cross-relation defined on $\N \times \N$ by: :$\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$ Let $\eqclass {x, y} {}$ denote the equivalence class of $\tuple {x, y}$ under $\boxtimes$. The operation $\oplus$ on these equivalence classes is well-defined, in the sense that: {{begin-eqn}} {{eqn | l = \eqclass {a_1, b_1} {}       | r = \eqclass {a_2, b_2} {}       | c =  }} {{eqn | l = \eqclass {c_1, d_1} {}       | r = \eqclass {c_2, d_2} {}       | c =  }} {{eqn | ll= \leadsto       | l = \eqclass {a_1, b_1} {} \oplus \eqclass {c_1, d_1} {}       | r = \eqclass {a_2, b_2} {} \oplus \eqclass {c_2, d_2} {}       | c =  }} {{end-eqn}}
Preimage of Serial Relation is Domain
Let $\mathcal R$ be a serial relation on $S$. Then the preimage of $\mathcal R$ is $S$ (the domain of $\mathcal R$).
Closed Form for Octagonal Numbers
The closed-form expression for the $n$th octagonal number is: :$O_n = n \left({3 n - 2}\right)$
Number of Distinct Deltahedra is Unlimited
There are an unlimited number of distinct deltahedra.
Set Intersection Preserves Subsets/Corollary/Proof 1
Let $A, B, S$ be sets. Then: :$A \subseteq B \implies A \cap S \subseteq B \cap S$
Characteristic of Increasing Mapping from Toset to Order Complete Toset
Let $\struct {S, \preceq}$ and $\struct {T, \preccurlyeq}$ be tosets. Let $T$ be order complete. Let $H \subseteq S$ be a subset of $S$. Let $f: H \to T$ be an increasing mapping from $H$ to $T$. Then: :$f$ has an extension to $S$ which is increasing : :for all $A \subseteq H$: if $A$ is bounded in $S$, then $f \sqbrk A$ is bounded in $T$ where $f \sqbrk A$ denotes the image set of $A$ under $f$.
A.E. Equal Positive Measurable Functions have Equal Integrals
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $f, g: X \to \overline \R_{\ge 0}$ be positive $\mu$-measurable functions. Suppose that $f = g$ almost everywhere. Then: :$\displaystyle \int f \rd \mu = \int g \rd \mu$
Characterization of Euler's Number by Inequality
Let $a$ be a (strictly) positive real number. Then: :$a = e \iff \forall x \in \R: a^x \ge x + 1$ where $e$ denotes Euler's number.
Limit of x to the x
Let $f: \R \to \R$ be defined on $\left [{0 \,.\,.\, \to} \right)$ with $f \left({x}\right) = x^x$. Then: :$\displaystyle \lim_{x \to 0^+} x^x = 1$ Equivalently, from the definition of power: :$\displaystyle \lim_{x \to 0^+} \exp \left({x \ln x}\right) = 1$
Schanuel's Conjecture Implies Transcendence of Log Pi
Let Schanuel's Conjecture be true. Then the logarithm of $\pi$ (pi): :$\ln \pi$ is transcendental.
Cauchy-Bunyakovsky-Schwarz Inequality/Lebesgue 2-Space
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $f, g: X \to \R$ be $\mu$-square integrable functions, that is $f, g \in \map {\LL^2} \mu$, Lebesgue $2$-space. Then: :$\displaystyle \int \size {f g} \rd \mu \le \norm f_2^2 \cdot \norm g_2^2$ where $\norm {\, \cdot \,}_2$ is the $2$-norm.
Preimage of Intersection under Relation/Family of Sets
Let $S$ and $T$ be sets. Let $\family {T_i}_{i \mathop \in I}$ be a family of subsets of $T$. Let $\RR \subseteq S \times T$ be a relation. Then: :$\ds \RR^{-1} \sqbrk {\bigcap_{i \mathop \in I} T_i} \subseteq \bigcap_{i \mathop \in I} \RR^{-1} \sqbrk {T_i}$ where $\ds \bigcap_{i \mathop \in I} T_i$ denotes the intersection of $\family {T_i}_{i \mathop \in I}$.
Image of Intersection under Injection/Family of Sets
Let $S$ and $T$ be sets. Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$. Let $f: S \to T$ be a mapping. Then: :$\displaystyle f \sqbrk {\bigcap_{i \mathop \in I} S_i} = \bigcap_{i \mathop \in I} f \sqbrk {S_i}$  $f$ is an injection.
Lines through Center Square of Order 3 Magic Square are in Arithmetic Sequence
Consider the order 3 magic square: {{:Magic Square/Examples/Order 3}} Each of the lines through the center cell contain $3$ integers in arithmetic sequence.
Omega Constant is Transcendental
The omega constant is transcendental.
Divisibility by Power of 10
Let $r \in \Z_{\ge 1}$ be a strictly positive integer. An integer $N$ expressed in decimal notation is divisible by $10^r$  the last $r$ digits of $N$ are all $0$. That is: :$N = \sqbrk {a_n \ldots a_2 a_1 a_0}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $10^r$  : :$a_0 + a_1 10 + a_2 10^2 + \cdots + a_r 10^r = 0$
Correspondence Theorem (Set Theory)
Let $S$ be a set. Let $\RR \subseteq S \times S$ be an equivalence relation on $S$. Let $\mathscr A$ be the set of partitions of $S$ associated with equivalence relations $\RR'$ on $S$ such that: :$\tuple {x, y} \in \RR \iff \tuple {x, y} \in \RR'$ Then there exists a bijection $\phi$ from $\mathscr A$ onto the set of partitions of the quotient set $S / \RR$.
Factorial as Product of Three Factorials
This general pattern can be used to find a factorial which is the product of three factorials: :$\left({\left({n!}\right)!}\right)! = n! \left({n! - 1}\right)! \left({\left({n!}\right)! - 1}\right)!$ while there are instances of factorials which do not fit that pattern.
Right Quasigroup if (1-3) Parastrophe of Magma is Magma
Let $\struct {S, \circ}$ be a magma. Let the $(1-3)$ parastrophe of $\struct {S, \circ}$ be a magma. Then $\struct {S, \circ}$ is a right quasigroup.
Left Quasigroup if (2-3) Parastrophe of Magma is Magma
Let $\struct {S, \circ}$ be a magma. Let the $\paren {2 - 3}$ parastrophe of $\struct {S, \circ}$ be a magma. Then $\struct {S, \circ}$ is a left quasigroup.
Product of Cardinals is Associative
Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be cardinals. Then: : $\mathbf a \left({\mathbf b \mathbf c}\right) = \left({\mathbf a \mathbf b}\right) \mathbf c$ where $\mathbf a \mathbf b$ denotes the product of $\mathbf a$ and $\mathbf b$.
Fundamental Theorem of Line Integrals
Let $\mathcal C$ be a smooth curve given by the vector function $\mathbf r \left({t}\right)$ for $a \le t \le b$. Let $f$ be a differentiable function of two or three variables whose gradient vector $\nabla f$ is continuous on $\mathcal C$. Then: :$\displaystyle \int_\mathcal C \nabla f \cdot d \mathbf r = f \left({\mathbf r \left({b}\right)}\right) - f \left({\mathbf r \left({a}\right)}\right)$
Abelian Group Induces Commutative B-Algebra
Let $\left({G, \circ}\right)$ be an abelian group whose identity element is $e$. Let $*$ be the binary operation on $G$ defined as: :$\forall a, b \in G: a * b = a \circ b^{-1}$ where $b^{-1}$ is the inverse element of $b$ under the operation $\circ$. Then the algebraic structure $\left({G, *}\right)$ is a commutative $B$-algebra. That is: :$\forall a, b \in G: a * \left({0 * b}\right) = b * \left({0 * a}\right)$
Sum of Cardinals is Associative
Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be cardinals. Then: : $\mathbf a + \left({\mathbf b + \mathbf c}\right) = \left({\mathbf a + \mathbf b}\right) + \mathbf c$ where $\mathbf a + \mathbf b$ denotes the sum of $\mathbf a$ and $\mathbf b$.
Lucas Number 2n in terms of Square of Lucas Number n
Let $L_n$ denote the $n$th Lucas number. Then: :$L_{2 n} = {L_n}^2 + 2 \left({-1}\right)^n$
Natural Numbers as Cardinals
The natural numbers $\N = \set {0, 1, 2, 3, \ldots}$ can be defined as the set of cardinals.
Young's Inequality for Convolutions
Let $p, q, r \in \R_{\ge 1}$ satisfy: :$1 + \dfrac 1 r = \dfrac 1 p + \dfrac 1 q$ Let $L^p \left({\R^n}\right)$, $L^q \left({\R^n}\right)$, and $L^r \left({\R^n}\right)$ be Lebesgue spaces with seminorms $\left\Vert{\cdot}\right\Vert_p$, $\left\Vert{\cdot}\right\Vert_q$, and $\left\Vert{\cdot}\right\Vert_r$ respectively. Let $f \in L^p \left({\R^n}\right)$ and $g \in L^q \left({\R^n}\right)$. Then the convolution $f * g$ is in $L^r \left({\R^n}\right)$ and the following inequality is satisfied: :$\left\Vert{f * g}\right\Vert_r \le \left\Vert{f}\right\Vert_p \cdot \left\Vert{g}\right\Vert_q$
Conditions for C^1 Smooth Solution of Euler's Equation to have Second Derivative
Let $\map y x:\R \to \R$ be a real function. Let $\map F {x, y, y'}:\R^3 \to \R$ be a real function. Suppose $\map F {x, y, y'}$ has continuous first and second derivatives {{WRT|Differentiation}} all its arguments. Suppose $y$ has a continuous first derivative and satisfies Euler's equation:  :$F_y - \dfrac \d {\d x} F_{y'} = 0$ Suppose: :$\map {F_{y' y'} } {x, \map y x, \map y x'} \ne 0$ Then $\map y x$ has continuous second derivatives.
If Double Integral of a(x, y)h(x, y) vanishes for any C^2 h(x, y) then C^0 a(x, y) vanishes
Let $\alpha \left({x, y}\right)$, $h \left({x, y}\right)$ be functions in $\R$. Let $\alpha \in C^0$ in a closed region $R$ whose boundary is $\Gamma$. Let $h \in C^2$ in $R$ and $h = 0$ on $\Gamma$. Let: : $\displaystyle \int \int_R \alpha \left({x, y}\right) h \left({x, y}\right) \rd x \rd y = 0$ Then $\alpha \left({x, y}\right)$ vanishes everywhere in $R$.
Simple Variable End Point Problem
Let $y$ and $F$ be mappings. {{explain|Define their domain and codomain}} Suppose the endpoints of $y$ lie on two given vertical lines $x = a$ and $x = b$. Suppose $J$ is a functional of the form :$(1): \quad J \sqbrk y = \displaystyle \int_a^b \map F {x, y, y'} \rd x$ and has an extremum for a certain function $\hat y$. Then $y$ satisfies the system of equations :$\begin {cases} F_y - \dfrac \d {\d x} F_{y'} = 0 \\ \bigvalueat {F_{y'} } {x \mathop = a} = 0 \\ \bigvalueat {F_{y'} } {x \mathop = b} = 0 \end {cases}$
Ordered Set of Closure Operators and Dual Ordered Set of Closure Systems are Isomorphic
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Then $\operatorname{Closure}\left({L}\right)$ and $\operatorname{ClSystems}\left({L}\right)^{-1}$ are order isomorphic where :$\operatorname{Closure}\left({L}\right)$ denotes the ordered set of closure operators of $L$, :$\operatorname{ClSystems}\left({L}\right)$ denotes the ordered set of closure systems oj $L$, :$\operatorname{ClSystems}\left({L}\right)^{-1}$ denotes the dual to $\operatorname{ClSystems}\left({L}\right)$.
Euler's Equation for Vanishing Variation is Invariant under Coordinate Transformations
Euler's Equation for Vanishing Variation is invariant under coordinate transformations.
Minimal Infinite Successor Set is Limit Ordinal
Let $\omega$ denote the minimal infinite successor set. Then $\omega$ is a limit ordinal.
Ordinals Isomorphic to the Same Well-Ordered Set
Let $A$ and $B$ be ordinals. Let $\left({\prec, S}\right)$ be a strict well-ordering. Let $\left({\in, A}\right)$ and $\left({\prec, S}\right)$ be order isomorphic. Let $\left({\in, B}\right)$ and $\left({\prec, S}\right)$ be order isomorphic. Then: : $A = B$
Topological Group is T1 iff T2
Let $G$ be a topological group. Then $G$ is a $T_1$ space  $G$ is Hausdorff.
Logarithm of Infinite Product of Complex Numbers
Let $\sequence {z_n}$ be a sequence of nonzero complex numbers.  :$(1): \quad$ The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty z_n$ converges to $z \in \C_{\ne 0}$. :$(2): \quad$ The series $\displaystyle \sum_{n \mathop = 1}^\infty \log z_n$ converges to $\log z + 2 k \pi i$ for some integer $k \in \Z$.
Infinite Ordinal can be expressed Uniquely as Sum of Limit Ordinal plus Finite Ordinal
Let $x$ be an ordinal. Suppose $x$ satisfies $\omega \subseteq x$. Then $x$ has a unique representation as $\paren {y + z}$ where $y$ is a limit ordinal and $z$ is a finite ordinal.
Legs of Pythagorean Triangle used as Generator for another Pythagorean Triangle
Let $a$ and $b$ be the legs of a Pythagorean triangle $P_1$. Let $\tuple {a, b}$ be used as the generator for a new Pythagorean triangle $P_2$. Then the hypotenuse of $P_2$ is the square of the hypotenuse of $P_1$.
Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation
Let $\Phi = \map {\Phi} {x, \family {y_i}_{1 \mathop \le i \mathop \le n}, \family {p_i}_{1 \mathop \le i \mathop \le n} }$ be a real function. Let $H$ be Hamiltonian. Then a necessary and sufficient condition for $\Phi$ to be the first integral of Euler's Equations is :$\dfrac {\partial \Phi} {\partial x} + \sqbrk{\Phi, H} = 0$
Homotopic Paths Implies Homotopic Composition
Let $T = \left({S, \tau}\right)$ be a topological space. Let $f_1, f_2, g_1, g_2: \left[{0 \,.\,.\, 1}\right] \to S$ be paths in $T$. Let $f_1$ be homotopic to $f_2$ and $g_1$ be homotopic to $g_2$. Then the concatenated paths $f_1 * g_1$ and $f_2 * g_2$ are homotopic.
Product of Two Triangular Numbers to make Square
Let $T_n$ be a triangular number. Then there is an infinite number of $m \in \Z_{>0}$ such that $T_n \times T_m$ is a square number.
Numbers of form 31 x 16^n are sum of 16 Powers of 4
Let $m \in \Z$ be an integer of the form $31 \times 16^n$ for $n \in \Z_{\ge 0}$. Then in order express $m$ as the sum of powers of $4$, you need $16$ of them.
Element of Ordered Set of Topology is Dense iff is Everywhere Dense
Let $T = \struct {S, \tau}$ be a topological space. Let $P = \struct {\tau, \preceq}$ be an ordered set where $\mathord\preceq = \mathord\subseteq \cap \paren {\tau \times \tau}$ Let $A \in \tau$. Then $A$ is a dense element in $P$  $A$ is everywhere dense.
Element Commutes with Product of Commuting Elements/General Theorem
Let $(S,\circ)$ be a semigroup. Let $\left \langle {a_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ be a sequence of terms of $S$. Let $b \in S$. If $b$ commutes with $a_k$ for each $k \in \left[{1 \,.\,.\, n}\right]$, then $b$ commutes with $a_1 \circ \cdots \circ a_n$.
Set is Small Class
Let $x$ be a set. Then $x$ is a small class.
Group Induced by B-Algebra Induced by Group
Let $\left({S, \circ}\right)$ be a group. Let $\left({S, *}\right)$ be the $B$-algebra described on Group Induces $B$-Algebra. Let $\left({S, \circ'}\right)$ be the group described on $B$-Algebra Induces Group. Then $\left({S, \circ'}\right) = \left({S, \circ}\right)$.
Category of Ordered Sets is Category
Let $\mathbf{OrdSet}$ be the category of ordered sets. Then $\mathbf{OrdSet}$ is a metacategory.
Magic Hexagon of Order 3 is Unique
Apart from the trivial order $1$ magic hexagon, there exists only one magic hexagon: the order $3$ magic hexagon: {{:Definition:Order 3 Magic Hexagon}}
Universal Class is Proper
Let $V$ denote the universal class. Then $V$ is a proper class.
Squares Ending in 5 Occurrences of 2-Digit Pattern
Let $n$ be a square number whose decimal representation ends in the pattern $\mathtt {xyxyxyxyxy}$. Then $\mathtt {xy}$ is one of: :$21, 29, 61, 69, 84$ The smallest examples of such numbers are: {{begin-eqn}} {{eqn | l = 508 \, 853 \, 989^2       | r = \phantom {0 \,} 258 \, 932 \, 38 \mathbf {2 \, 121 \, 212 \, 121} }} {{eqn | l = 162 \, 459 \, 327^2       | r = \phantom {0 \, 0} 26 \, 393 \, 03 \mathbf {2 \, 929 \, 292 \, 929} }} {{eqn | l = 1 \, 318 \, 820 \, 881^2       | r = 1 \, 739 \, 288 \, 51 \mathbf {6 \, 161 \, 616 \, 161} }} {{eqn | l = 541 \, 713 \, 187^2       | r = \phantom {0 \,} 293 \, 453 \, 17 \mathbf {6 \, 969 \, 696 \, 969} }} {{eqn | l = 509 \, 895 \, 478^2       | r = \phantom {0 \,} 259 \, 993 \, 39 \mathbf {8 \, 484 \, 848 \, 484} }} {{end-eqn}}