hcju/proofwikips huggingface dataset:
Infima Preserving Mapping on Filters is Increasing
Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be ordered sets. Let $f: S \to T$ be a mapping. For every filter $F$ in $\struct {S, \preceq}$, let $f$ preserve the infimum on $F$. Then $f$ is increasing.
Infimum of Upper Closure of Set
Let $\left({S, \preceq}\right)$ be an ordered set. Let $T \subseteq S$. Let $U = T^\succeq$ be the upper closure of $T$ in $S$. Let $s \in S$. Then $s$ is the infimum of $T$  it is the infimum of $U$.
Cavalieri's Principle/Extension
Let two solid figures $S_1$ and $S_2$ have equal height. Let the areas of the sections made by planes parallel to their bases and at equal distances from the bases always have the same ratio. Then the volumes of $S_1$ and $S_2$ are in that same ratio.
Cavalieri's Principle
Let two solid figures $S_1$ and $S_2$ have equal height. Let sections made by planes parallel to their bases and at equal distances from the bases always have equal area. Then the volumes of $S_1$ and $S_2$ are equal.
Volume of Cone
Let $K$ be a cone whose base is of area $A$ and whose height is $h$. Then the volume of $K$ is given by: :$V_K = \dfrac {A h} 3$
Volume of Cylinder
The volume $V_C$ of a cylinder whose bases are circles of radius $r$ and whose height is $h$ is given by the formula: :$V_C = \pi r^2 h$
Slope of Tangent to Cycloid
Let $C$ be a cycloid generated by the equations: :$x = a \paren {\theta - \sin \theta}$ :$y = a \paren {1 - \cos \theta}$ The slope of the tangent to $C$ at the point $\tuple {x, y}$ is given by: :$\dfrac {\d y} {\d x} = \cot \dfrac \theta 2$
Upper Closure of Singleton
Let $\left({S, \preceq}\right)$ be an ordered set. Let $s$ be an element of $S$. Then: :$\left\{ {s}\right\}^\succeq = s^\succeq$ where: :$\left\{ {s}\right\}^\succeq$ denotes the upper closure of $\left\{ {s}\right\}$ :$s^\succeq$ denotes the upper closure of $s$
Lower Closure of Singleton
Let $\left({S, \preceq}\right)$ be an ordered set. Let $s$ be an element of $S$. Then: : $\left\{ {s}\right\}^\preceq = s^\preceq$ where: :$\left\{ {s}\right\}^\preceq$ denotes the lower closure of $\left\{ {s}\right\}$ :$s^\preceq$ denotes the lower closure of $s$.
Upper Closure of Element is Filter
Let $\struct {S, \preceq}$ be an ordered set. Let $s$ be an element of $S$. Then: :$s^\succeq$ is a filter in $\struct {S, \preceq}$ where $s^\succeq$ denotes the upper closure of $s$.
Length of Logarithmic Spiral
Consider a logarithmic spiral $S$ given by the equation: :$r = a e^{b \theta}$ Construct a tangent to $S$ at the point $Q = \left({a, 0}\right)$. Let the tangent cross the $y$-axis at $P$. Then the length of $PQ$ equals the total length of $S$ from $P$ inwards to the origin.
Length of Logarithmic Spiral/Historical Note
The length of a logarithmic spiral was first found by {{AuthorRef|Evangelista Torricelli}} in $1645$. This was the first time anybody had found the length of a non-straight-line curve for anything other than a circle. Before this had been done, few people could accept that this was possible to do. For example, {{AuthorRef|René Descartes}} had stated in his {{BookLink|La Géométrie|René Descartes}} in $1637$: :''Geometry should not include lines that are like strings, in that they are sometimes straight and sometimes curved, since the ratios between straight and curved lines are not known, and I believe cannot be discovered by human minds.'' {{AuthorRef|Galileo}}'s response was: :'' Who is so blind as not to see that, if there are two equal straight lines, one of which is then bent into a curve, that curve will be equal to the straight line?''
Infimum of Upper Closure of Element
Let $\struct {S, \preceq}$ be an ordered set. Let $s$ be an element of $S$. Then: :$\map \inf {s^\succeq} = s$ where $s^\succeq$ denotes the upper closure of $s$.
Volume of Gabriel's Horn
Consider Gabriel's horn, the solid of revolution formed by rotating about the $x$-axis the curve: :$y = \dfrac 1 x$ Consider the volume $V$ of the space enclosed by the planes $x = 1$, $x = a$ and the portion of Gabriel's horn where $1 \le x \le a$. Then: :$V = \pi \left({1 - \dfrac 1 a}\right)$
Pascal's Theorem
Let $ABCDEF$ be a hexagon whose $6$ vertices lie on a conic section and whose opposite sides are not parallel. Then the points of intersection of the opposite sides, when produced as necessary, all lie on a straight line.
Lower Closure of Element is Ideal
Let $\left({S, \preceq}\right)$ ne an ordered set. Let $s$ be an element of $S$. Then $s^\preceq$ is ideal in $\left({S, \preceq}\right)$ where $s^\preceq$ denotes the lower closure of $s$.
Supremum of Lower Closure of Element
Let $\left({S. \preceq}\right)$ be an ordered set. Let $s$ be an element of $S$. Then: :$\sup \left({s^\preceq}\right) = s$ where $s^\preceq$ denotes the lower closure of $s$.
Upper Closure is Decreasing
Let $\struct {S, \preceq}$ be an ordered set. Let $x, y$ be elements of $S$ such that :$x \preceq y$ then $y^\succeq \subseteq x^\succeq$ where $y^\succeq$ denotes the upper closure of $y$.
Lower Closure is Increasing
Let $\struct {S, \preceq}$ be an ordered set. Let $x, y$ be elements of $S$ such that :$x \preceq y$ then $x^\preceq \subseteq y^\preceq$ where $y^\preceq$ denotes the lower closure of $y$.
Suprema Preserving Mapping on Ideals is Increasing
Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets. Let $f: S \to T$ be a mapping. For every ideal $I$ in $\left({S, \preceq}\right)$, let $f$ preserve the supremum on $I$. Then $f$ is increasing.
Imaginary Unit to Power of Itself
:$i^i = e^{-\pi / 2}$ where $i$ is the imaginary unit. Its decimal expansion is: :$0 \cdotp 20787 \, 95763 \, 50761 \, 90854 \, 6955 \ldots$ {{OEIS|A049006}}
Imaginary Unit to Power of Itself/Complete
:$i^i = \set {\exp \paren {\dfrac {4 k + 3} 2 \pi}: k \in \Z}$ where $i$ is the imaginary unit.
Riemann Zeta Function of 4
The Riemann zeta function of $4$ is given by: {{begin-eqn}} {{eqn | l = \map \zeta 4       | r = \dfrac 1 {1^4} +  \dfrac 1 {2^4} +  \dfrac 1 {3^4} +  \dfrac 1 {4^4} + \cdots       | c =  }} {{eqn | r = \dfrac {\pi^4} {90}       | c =  }} {{eqn | o = \approx       | r = 1 \cdotp 08232 \, 3 \ldots       | c =  }} {{end-eqn}}
Riemann Zeta Function of 6
The Riemann zeta function of $6$ is given by: {{begin-eqn}} {{eqn | l = \map \zeta 6       | r = \dfrac 1 {1^6} +  \dfrac 1 {2^6} +  \dfrac 1 {3^6} +  \dfrac 1 {4^6} + \cdots       | c =  }} {{eqn | r = \dfrac {\pi^6} {945}       | c =  }} {{eqn | o = \approx       | r = 1 \cdotp 01734 \, 3 \ldots       | c =  }} {{end-eqn}}
Infima Preserving Mapping on Filters Preserves Filtered Infima
Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be ordered sets. Let $f: S \to T$ be a mapping. For every filter $F$ in $\struct {S, \preceq}$, let $f$ preserve the infimum on $F$. Then $f$ preserves filtered infima.
Approximate Formula for Number of Partitions
The number of partitions $\map p n$ of a (strictly) positive integer $n$ is given by the approximation: :$\map p n \approx \dfrac {e^{\pi \sqrt {2 n / 3} } } {4 n \sqrt 3}$
Number of Partitions as Coefficient of Power Series
The number of partitions $\map p n$ of a (strictly) positive integer $n$ is equal to the coefficient of $x^n$ when the expression: :$\map f n = \dfrac 1 {\paren {1 - x} \paren {1 - x^2} \paren {1 - x^3} \cdots}$ is expanded into a power series. That is: :$\map f n = 1 + \map p 1 x + \map p 2 x^2 + \map p 3 x^3 + \cdots$ or: :$\displaystyle \sum_{n \mathop \in \Z_{\ge 0} } \map p n q^n = \prod_{j \mathop \in \Z_{>0} } \dfrac 1 {1 - q^j}$ where $\map p 0 := 1$.
Suprema Preserving Mapping on Ideals Preserves Directed Suprema
Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets. Let $f: S \to T$ be a mapping. Let every filter $F$ in $\left({S, \preceq}\right)$, $f$ preserve the infimum on $F$. Then $f$ preserves directed suprema.
Euler's Equations of Motion for Rotation of Rigid Body
Let a rigid body $B$ rotate about an axis $A$ which is fixed in relation to $B$ and parallel to the principal axis of inertia of $B$. Then the rotation of $B$ about $A$ is described by: :$\mathbf I \cdot \dot {\boldsymbol \omega} + \boldsymbol \omega \times \left({\mathbf I \cdot \boldsymbol\omega}\right) = \mathbf M$ where: :$\mathbf M$ is the applied torque applied to $B$ about $A$ :$\mathbf I$ is the moment of inertia of $B$ with respect to $A$ :$\boldsymbol \omega$ is the angular velocity about $A$.
Euler's Hydrodynamical Equation for Flow of Ideal Incompressible Fluid
:$\begin{cases} \dfrac {D \mathbf u} {D t} & = & - \nabla w + \mathbf g \\ \nabla \cdot \mathbf u & = & 0 \end{cases}$ where: :$\mathbf u$ denotes the flow velocity vector, with components in an $N$-dimensional space $u_1, u_2, \dots, u_N$ :$\dfrac D {D t}$ denotes the material derivative in time :$\cdot$ denotes the dot product :$\nabla$ denotes the nabla operator, used to represent the specific thermodynamic work gradient (first equation), and the flow velocity divergence (second equation) :$\mathbf u \cdot \nabla$ is the convective derivative :$w$ is the thermodynamic work per unit mass, the internal source term :$\mathbf g$ denotes body acceleration per unit mass acting on the continuum. {{stub|Absolutely no idea as to how best to present this. Leaving it as a stub for now.}} {{Namedfor|Leonhard Paul Euler|cat = Euler}}
Euler Buckling Formula
:$F = \dfrac {\pi^2 E I} {\paren {K L}^2}$ where: :$F$ = maximum or critical force (vertical load on column) :$E$ = modulus of elasticity :$I$ = area moment of inertia of the cross section of the rod :$L$ = unsupported length of column :$K$ = column effective length factor, whose value depends on the conditions of end support of the column, as follows: ::For both ends pinned (hinged, free to rotate), $K = 1.0$ ::For both ends fixed, $K = 0.50$ ::For one end fixed and the other end pinned, $K \approx 0.699$ ::For one end fixed and the other end free to move laterally, $K = 2.0$ :$K L$ is the effective length of the column
Mellin Transform of Heaviside Step Function/Lemma
Let $t \in \R$. Let $s \in \C$ with $\map \Re s < 0$. Then: :$\displaystyle \lim_{t \mathop \to +\infty} t^s = 0$
Up-Complete Lower Bounded Join Semilattice is Complete
Let $\left({S, \preceq}\right)$ be an up-complete lower bounded join semillattice. Then $\left({S, \preceq}\right)$ is complete.
Gregory Series
For $-\dfrac \pi 4 \le \theta \le \dfrac \pi 4$: :$\theta = \tan \theta - \dfrac 1 3 \tan^3 \theta + \dfrac 1 5 \tan^5 \theta - \ldots$
Differential Form of Arc Length on Curved Surface
The first fundamental form for the element of arc length on a curved surface is given by: :$\d s^2 = E \rd u^2 + 2 F \d u \rd v + G \rd v^2$ where $E$, $F$ and $G$ are the coefficients of the first fundamental form.
Lattice is Complete iff it Admits All Suprema
Let $\left({S, \preceq}\right)$ be an ordered set. Then $\left({S, \preceq}\right)$ is a complete lattice  :$\forall X \subseteq S: X$ admits a supremum.
Theorema Egregium
The Gaussian curvature of a surface does not change if one bends the surface without stretching it. That is, Gaussian curvature can be determined entirely by measuring angles, distances and their rates on the surface itself, without further reference to the particular way in which the surface is embedded in the ambient $3$-dimensional Euclidean space. Thus the Gaussian curvature is an intrinsic invariant of a surface.
Gauss-Bonnet Theorem
Let $M$ be a compact $2$-dimensional Riemannian manifold with boundary $\partial M$. Let $\Kappa$ be the Gaussian curvature of $M$. Let $k_g$ be the geodesic curvature of $\partial M$. Then : :$\displaystyle \int_M \kappa \, \mathrm d A + \int_{\partial M} k_g \, \mathrm d s = 2 \pi \chi\left({M}\right)$ where: :$\mathrm d A$ is the element of area of the surface :$\mathrm d s$ is the line element along $\partial M$ :$\chi\left({M}\right)$ is the Euler characteristic of $M$.
Unique Factorization Theorem for Gaussian Integers
For every Gaussian integer $n$ such that $\left\vert{n}\right\vert > 1$, $n$ can be expressed as the product of one or more Gaussian primes, uniquely up to the order in which they appear.
Dirichlet's Principle (Harmonic Functions)
Let the function $\map u x$ be the particular solution to Poisson's equation: :$\Delta u + f = 0$ on a domain $\Omega$ of $\R^n$ with boundary condition: :$u = g$ on $\partial \Omega$ Then $u$ can be obtained as the minimizer of the Dirichlet's energy: :$\displaystyle E \sqbrk {\map v x} = \int_\Omega \paren {\frac 1 2 \cmod {\nabla v}^2 - v f} \rd x$ amongst all twice differentiable functions $v$ such that $v = g$ on $\partial \Omega$ . This result holds provided that there exists at least one function which makes the Dirichlet integral finite.
Galois Connection is Expressed by Minimum
Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be ordered sets. Let $g: S \to T$, $d: T \to S$ be mappings. Then $\tuple {g, d}$ is a Galois connection : :$g$ is an increasing mapping and ::$\forall t \in T: \map d t = \min \set {g^{-1} \sqbrk {t^\succsim} }$ where :$\min$ denotes the minimum :$g^{-1} \sqbrk {t^\succsim}$ denotes the image of $t^\succsim$ under relation $g^{-1}$ :$t^\succsim$ denotes the upper closure of $t$
Closed Form for Pentagonal Numbers
The closed-form expression for the $n$th pentagonal number is: :$P_n = \dfrac {n \paren {3 n - 1} } 2$
Closed Form for Hexagonal Numbers
The closed-form expression for the $n$th hexagonal number is: :$H_n = n \paren {2 n - 1}$
Sum of Even Integers is Even
The sum of any finite number of even integers is itself even.
Galois Connection is Expressed by Maximum
Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be ordered sets. Let $g: S \to T$, $d: T \to S$ be mappings. Then $\tuple {g, d}$ is a Galois connection  :$d$ is an increasing mapping and: ::$\forall s \in S: \map g s = \map \max {d^{-1} \sqbrk {s^\preceq} }$ where: :$\max$ denotes the maximum :$d^{-1} \sqbrk {s^\preceq}$ denotes the image of $s^\preceq$ under relation $d^{-1}$ :$s^\preceq$ denotes the lower closure of $t$
Compass and Straightedge Construction for Regular Heptagon does not exist
There exists no compass and straightedge construction for the regular heptagon.
Integer to Rational Power is Irrational iff not Integer or Reciprocal
Let $m$ be a rational number. Let $n$ be a positive integer. Then $n^m$ is an irrational number  $n^{\size m}$ is not an integer.
Squaring the Circle by Archimedean Spiral
Let $C$ be a circle. It is possible to construct a square of the same area as $C$ using an Archimedean spiral.
Surface Area of Sphere
The surface area $A$ of a sphere whose radius $a$ is given by: :$A = 4 \pi a^2$
Upper Bound for Number of Grains of Sand to fill Universe
The number of grains of sand that would fill the Universe, to the extent it was believed to exist in the days of {{AuthorRef|Archimedes of Syracuse|Archimedes}}, is bounded above by $10^{63}$.
Lower Adjoint Preserves All Suprema
Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets. Let $d: T \to S$ be an lower adjoint of Galois connection. Then $d$ preserves all suprema.
Oscillation at Point (Infimum) equals Oscillation at Point (Limit)
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 $x$ as defined by: :$\map {\omega_f} x = \inf \set {\map {\omega_f} I: I \in N_x}$ where $\map {\omega_f} I$ is the oscillation of $f$ on a real set $I$: :$\map {\omega_f} I = \sup \set {\size {\map f y - \map f z}: y, z \in I \cap D}$ Let $\map {\omega^L_f} x$ be the oscillation of $f$ at $x$ as defined by: :$\map {\omega^L_f} x = \displaystyle \lim_{h \mathop \to 0^+} \map {\omega_f} {\openint {x - h} {x + h} }$ Then: :$\map {\omega_f} x \in \R$  $\map {\omega^L_f} x \in \R$ and, if $\map {\omega_f} x$ and $\map {\omega^L_f} x$ exist as real numbers: :$\map {\omega_f} x = \map {\omega^L_f} x$
All Infima Preserving Mapping is Upper Adjoint of Galois Connection
Let $\left({S, \preceq}\right)$ be a complete lattice. Let $\left({T, \precsim}\right)$ be an ordered set. Let $g: S \to T$ be all infima preserving mapping. Then there exists a mapping $d: T \to S$ such that $\left({g, d}\right)$ is Galois connection and :$\forall t \in T: d\left({t}\right) = \min\left({g^{-1}\left[{t^\succsim}\right]}\right)$ where :$\min$ denotes the minimum :$g^{-1}\left[{t^\succsim}\right]$ denotes the image of $t^\succsim$ under relation $g^{-1}$ :$t^\succsim$ denotes the upper closure of $t$
Binomial Theorem/General Binomial Theorem/Historical Note
The General Binomial Theorem was first conceived by {{AuthorRef|Isaac Newton}} during the years $1665$ to $1667$ when he was living in his home in Woolsthorpe. He announced the result formally, in letters to [https://en.wikipedia.org/wiki/Henry_Oldenburg Henry Oldenburg] on $13$th June $1676$ and $24$th October $1676$ but did not provide a proper proof (at that time the need for the appropriate level of rigor had not been recognised). {{AuthorRef|Leonhard Paul Euler}} made an incomplete attempt in $1774$, but the full proof had to wait for {{AuthorRef|Carl Friedrich Gauss}} to provide it in $1812$. This was, in fact, the first time anything about infinite summations was proved adequately.
Image of Inverse Image
Let $S, T$ be sets. Let $f: S \to T$ be a mapping. Let $X$ be a subset of $T$. Then: :$f\left[{f^{-1}\left[{X}\right]}\right] \subseteq X$ where: :$f^{-1}\left[{X}\right]$ denotes the image of $X$ under the relation $f^{-1}$.
All Suprema Preserving Mapping is Lower Adjoint of Galois Connection
Let $\struct {S, \preceq}$ be an ordered set. Let $\struct {T, \precsim}$ be a complete lattice. Let $d: T \to S$ be all suprema preserving mapping. Then there exists a mapping $g: S \to T$ such that $\left({g, d}\right)$ is Galois connection and :$\forall s \in S: \map g s = \map \max {d^{-1} \sqbrk {s^\preceq} }$ where :$\min$ denotes the maximum :$d^{-1} \sqbrk {s^\preceq}$ denotes the image of $s^\preceq$ under relation $d^{-1}$ :$s^\preceq$ denotes the lower closure of $t$
Galois Connection Implies Order on Mappings
Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be ordered sets. Let $g: S \to T$ and $d: T \to S$ be mappings such that :$\tuple {g, d}$ is Galois connection. Then $d \circ g \preceq I_S$ and $I_T \precsim g \circ d$ where :$\preceq, \precsim$ denote the orderings on mappings, :$I_S$ denotes the identity mapping of $S$
Primitive of exp (-x^2) has no Solution in Elementary Functions
The primitive: :$\displaystyle \int \map \exp {-x^2} \rd x$ cannot be expressed in terms of a finite number of elementary functions.
Primitive of exp x over x has no Solution in Elementary Functions
The primitive: :$\displaystyle \int \frac {e^{a x} \, \mathrm d x} x$ cannot be expressed in terms of a finite number of elementary functions.
Primitive of Sine of x over x has no Solution in Elementary Functions
The primitive: :$\displaystyle \int \frac {\sin x \, \mathrm d x} x$ cannot be expressed in terms of a finite number of elementary functions.
Primitive of Reciprocal of Logarithm of x has no Solution in Elementary Functions
The primitive: :$\displaystyle \int \frac {\mathrm d x} {\ln x}$ cannot be expressed in terms of a finite number of elementary functions.
Pi is Transcendental
$\pi$ (pi) is transcendental.
Solution of Qunitic Equation using Elliptic Functions
A quintic equation can be solved using elliptic functions.
Riemann Surface is Path-Connected
A Riemann surface is path-connected.
Properties of Riemann Surface
This page gathers some results about Riemann surfaces.
Radó's Theorem (Riemann Surfaces)
A Riemann surface is second countable.
Conformal Isomorphism of Universal Cover of Riemann Surface
The universal cover of a Riemann surface is conformally isomorphic to either: :the Riemann sphere :the complex plane or :the unit disk.
Riemann Surface is Metrizable
A Riemann surface is metrizable.
Riemann Surface admits Metric of Constant Curvature
A Riemann surface admits a metric of constant curvature.
Riemann Sphere is only Elliptic Riemann Surface
The Riemann sphere is the only elliptic Riemann surface (up to conformal isomorphism).
Parabolic Riemann Surface is Plane, Punctured Plane or Torus
A parabolic Riemann surface is conformally isomorphic to either: :the complex plane :the punctured complex plane $\C \setminus \set 0$ or: :a torus.
Riemann's Rearrangement Theorem
Let $S$ be an infinite series which is conditionally convergent. Then its terms can be arranged in a permutation so that the new series converges to any given value, or diverges.
Riemann-Christoffel Tensor in Two Dimensions is Gaussian Curvature
Let $M$ be a Riemannian manifold of dimension $2$. Then the Riemann-Christoffel tensor on $M$ reduces to the Gaussian curvature on $M$.
Riemannian Manifold has Zero Gaussian Curvature iff Euclidean
Let $M$ be a Riemannian manifold of dimension $2$. The Gaussian curvature on $M$ is zero  the Riemannian metric on $\mathcal M$ is the same as the Euclidean metric.
Particle on Curved Surface under no Force moves along Geodesic
Consider a particle $P$ which is constrained to move on a curved surface $C$. Let $P$ be such that no force acts upon it. Then $P$ moves along a geodesic.
Sum of Reciprocals of Powers as Euler Product
Let $\zeta$ be the Riemann zeta function. Let $s \in \C$ be a complex number with real part $\sigma > 1$. Then: :$\displaystyle \map \zeta s = \prod_{\text {$p$ prime} } \frac 1 {1 - p^{-s} }$ where the infinite product runs over the prime numbers.
Shift Mapping is Lower Adjoint iff Appropriate Maxima Exist
Let $\left({S, \preceq}\right)$ be a meet semilattice. Then the following two conditions are equivalent: $(1): \quad \forall x \in S, f:S \to S: \left({\forall s \in S: f\left({s}\right) = x \wedge s}\right) \implies f$ is lower adjoint $(2): \quad \forall x, t \in S: \max \left\{ {s \in S: x \wedge s \preceq t}\right\}$ exists.
Brouwerian Lattice iff Shift Mapping is Lower Adjoint
Let $\left({S, \preceq}\right)$ be a lattice. Then $\left({S, \preceq}\right)$ is a Brouwerian lattice  :$\forall x \in S, f: S \to S: \left({\forall s \in S: f \left({s}\right) = x \wedge s}\right) \implies f$ is a lower adjoint
Brouwerian Lattice is Distributive
Let $\left({S, \preceq}\right)$ be a Brouwerian lattice. Then $\left({S, \preceq}\right)$ is a distributive lattice
Definite Integral of Reciprocal of Root of a Squared minus x Squared
:$\displaystyle \int_0^x \frac {\d t} {\sqrt{1 - t^2} } = \arcsin x$
Meromorphic Function is Elliptic iff Doubly Periodic
Let $f: \C \to \C$ be a meromorphic function.  Then $\map f z$ is an elliptic function  it is also doubly periodic.
Entire Function is Transcendental iff Power Series Expansion is Infinite
Let $f$ be an entire function. Then $f$ is '''transcendental'''  the power series expansion of $f$ has an infinite number of coefficients which are non-zero.
Entire Function is Transcendental iff not Complex Polynomial Function
Let $f$ be an entire function. Then $f$ is '''transcendental'''  it is not a complex polynomial function.
Complex Function is Entire iff it has Everywhere Convergent Power Series
Let $f: \C \to \C$ be a complex function. Then $f$ is an '''entire function'''  $f$ can be given by an everywhere convergent power series: :$\displaystyle \map f z = \sum_{n \mathop = 0}^\infty a_n z^n; \quad \lim_{n \mathop \to \infty} \sqrt [n] {\size {a_n} } = 0$
Brouwerian Lattice is Upper Bounded
Let $\struct {S, \vee, \wedge, \preceq}$ be a Brouwerian lattice. Then $S$ is upper bounded.
Extreme Value Theorem/Real Function
Let $f$ be continuous in a closed real interval $\left[{a \,.\,.\, b}\right]$. Then: :$\forall x \in \left[{a \,.\,.\, b}\right]: \exists x_s \in \left[{a \,.\,.\, b}\right]: f \left({x_s}\right) \le f \left({x}\right)$ :$\forall x \in \left[{a \,.\,.\, b}\right]: \exists x_n \in \left[{a \,.\,.\, b}\right]: f \left({x_n}\right) \ge f \left({x}\right)$
Galileo's Paradox
The natural numbers $\N$ are in one-to-one correspondence with their squares: That is, the mapping: :$\forall n \in \N: f: n \mapsto n^2$ is a bijection. Hence the set of square numbers is equinumerous to the set of natural numbers. That is, a set is equinumerous to one of its proper subsets.