hcju/proofwikips huggingface dataset:
Separation Axioms on Double Pointed Topology/T3.5 Axiom
Let $T_1 = \struct {S, \tau_S}$ be a topological space. Let $D = \struct {A, \set {\O, A} }$ be the indiscrete topology on an arbitrary doubleton $A = \set {a, b}$. Let $T = \struct {T_1 \times D, \tau}$ be the double pointed topological space on $T_1$. Then $T \times D$ is a $T_{3 \frac 1 2}$ space  $T$ is also a $T_{3 \frac 1 2}$ space.
Square of Vandermonde Matrix
The square of the Vandermonde matrix of order $n$: : $\mathbf V = \begin{bmatrix}   x_1 & x_2 & \cdots & x_n \\   x_1^2 & x_2^2 & \cdots & x_n^2 \\ \vdots & \vdots & \ddots & \vdots \\   x_1^n & x_2^n & \cdots & x_n^n \end{bmatrix}$ is symmetrical in $x_1, \ldots, x_n$. {{questionable|The case $n {{=}} 2$ left me clueless to what could possibly be intended here; only $\mathbf {V V}^T$ is trivially seen symmetric in the $x_n$, but this can hardly be called a square}}
Open and Closed Sets in Multiple Pointed Topology
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ be a finite set whose cardinality is greater than $1$. Let $D = \left({A, \left\{{\varnothing, A}\right\}}\right)$ be the indiscrete space on $A$. Let $T \times D$ be a multiple pointed topological space generated from $T$ and $D$. Let $H \subseteq T$. Then: : $H \times A$ is open in $T \times D$  $H$ is open in $T$ : $H \times A$ is closed in $T \times D$  $H$ is closed in $T$ : $H \times \varnothing$ is both open and closed in $T \times D$ Let $\varnothing \subset H_A \subset A$, that is, let $H_A$ be a proper subset of $A$. Then $H \times H_A$ is neither open nor closed in $T \times D$.
Basis for Box Topology
Let $\left \langle {\left({S_i, \tau_i}\right)} \right \rangle_{i \mathop \in I}$ be an $I$-indexed family of topological spaces. Let $S$ be the cartesian product of $\left \langle {S_i} \right \rangle_{i \mathop \in I}$. That is: :$\displaystyle S := \prod_{i \mathop \in I} S_i$ Define: :$\displaystyle \mathcal B := \left\{{\prod_{i \mathop \in I} U_i: \forall i \in I: U_i \in \tau_i}\right\}$ Then $\mathcal B$ is a synthetic basis on $S$.
Space with Open Point is Non-Meager
Let $T = \struct {S, \tau}$ be a topological space. Let $x \in S$ be an open point. Then $T$ is a non-meager space.
Continuous Mapping on Finite Union of Closed Sets
Let $T = \struct {X, \tau}$ and $S = \struct {Y,\sigma}$ be topological spaces. For all $i \in \set {1, 2, \ldots, n}$, let $C_i$ be closed in $T$. Let $f: X \to Y$ be a mapping such that the restriction $f \restriction_{C_i}$ is continuous for all $i$. Then $f$ is continuous on $C = \displaystyle \bigcup_{i \mathop = 1}^n C_i$, that is, $f \restriction_C$ is continuous. If $\family {C_i}$ is infinite, the result does not necessarily hold.
Equivalence of Definitions of T2 Space
{{TFAE|def = T2 Space|view = $T_2$ (Hausdorff) space}} Let $T = \struct {S, \tau}$ be a topological space.
Equivalence of Definitions of T3 Space
{{TFAE|def = T3 Space|view = $T_3$ space}} Let $T = \struct {S, \tau}$ be a topological space.
Big Implies Saturated
Let $\mathcal M$ be an $\mathcal L$-structure. Let $\kappa$ be a cardinal. If $\mathcal M$ is $\kappa$-big, then it is $\kappa$-saturated.
Limit Points in T1 Space
Let $T = \struct {S, \tau}$ be a topological space which satisfies the $T_1$ (Fréchet) axiom. Let $H \subset S$ be any subset of $S$. Let $x \in H$. Then $x$ is a limit point of $H$  every neighborhood of $x$ contains infinitely many points of $H$.
Singleton Point is Isolated
Let $T = \left({S, \tau}\right)$ be a topological space. Let $x \in S$. Then $x$ is an isolated point of the singleton set $\left\{{x}\right\}$, but not necessarily an isolated point of $T$.
Tietze Extension Theorem
Let $T = \struct {S, \tau}$ be a topological space which is normal. Let $A \subseteq S$ be a closed set in $T$. Let $f: A \to \R$ be a continuous mapping from $A \subseteq S$ to the real number line under the usual (Euclidean) topology. Then there exists a continuous extension $g: S \to \R$, i.e. such that: :$\forall s \in A: \map f s = \map g s$
Hermitian Operators have Real Eigenvalues
Hermitian operations have real eigenvalues.
Hausdorff Space iff Diagonal Set on Product is Closed
Let $T = \struct{S, \tau}$ be a topological space. Let $\Delta_S$ be the diagonal set on $S$: :$\Delta_S = \set{\tuple{x, x} \in S \times S: x \in S}$ where $S \times S$ is the Cartesian product of $S$ with itself. Let $T^2 = \struct{S \times S, \TT}$ be the product space with Tychonoff topology $\TT$. Then $T$ is a $T_2$ (Hausdorff) space iff $\Delta_S$ is closed in $T^2$.
T0 Property is Hereditary
Let $T = \struct {S, \tau}$ be a topological space which is a $T_0$ (Kolmogorov) space. Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_0$ (Kolmogorov) space.
T1 Property is Hereditary
Let $T = \struct {S, \tau}$ be a topological space which is a $T_1$ (Fréchet) space. Let $T_H = \struct {H, \tau_H}$, where $\varnothing \subset H \subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_1$ (Fréchet) space.
T2 Property is Hereditary
Let $T = \struct {S, \tau}$ be a topological space which is a $T_2$ (Hausdorff) space. Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_2$ (Hausdorff) space. That is, the property of being a $T_2$ (Hausdorff) space is hereditary.
Completely Hausdorff Property is Hereditary
Let $T = \struct {S, \tau}$ be a topological space which is a $T_{2 \frac 1 2}$ (completely Hausdorff) space. Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space.
Closure in Subspace
Let $T = \struct {S, \tau}$ be a topological space. Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$. Let $U \subseteq H$ and let $\map {\cl_T} U$ be the closure of $U$ in $T$. Then $\map {\cl_T} U \cap H$ is the closure of $U$ in $T_H$. That is: :$\map {\cl_T} U \cap H = \map {\cl_H} U$ where $\map {\cl_H} U$ denotes the closure of $U$ in $T_H$.
T3 Property is Hereditary
Let $T = \struct {S, \tau}$ be a topological space which is a $T_3$ space. Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_3$ space.
T3 1/2 Property is Hereditary
Let $T = \struct {S, \tau}$ be a topological space which is a $T_{3 \frac 1 2}$ space. Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_{3 \frac 1 2}$ space.
T5 Property is Hereditary
Let $T = \struct {S, \tau}$ be a topological space which is a $T_5$ space. Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_5$ space.
T5 Space iff Every Subspace is T4
Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is a $T_5$ space  every subspace of $T$ is a $T_4$ space.
Product Space is T0 iff Factor Spaces are T0
Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $T = T_A \times T_B$ be the product space formed from $T_A$ and $T_B$. Then $T$ is a $T_0$ (Kolmogorov) space  $T_A$ and $T_B$ are themselves both $T_0$ (Kolmogorov) spaces.
Product Space is T1 iff Factor Spaces are T1
Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct {S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. Then $T$ is a $T_1$ (Fréchet) space  each of $\struct {S_\alpha, \tau_\alpha}$ is a $T_1$ (Fréchet) space.
Gödel's Incompleteness Theorems/First
Let $T$ be the set of theorems of some recursive set of sentences in the language of arithmetic such that $T$ contains minimal arithmetic. $T$ cannot be both consistent and complete.
Undecidability Theorem
Let $T$ be the set of theorems of some consistent theory in the language of arithmetic which contains minimal arithmetic $Q$. $T$ is not recursive.
Set of Gödel Numbers of Arithmetic Theorems Not Definable in Arithmetic
Let $T$ be the set of theorems of some consistent theory in the language of arithmetic which contains minimal arithmetic. The set of Gödel numbers of the theorems of $T$ is not definable in $T$.
Diagonal Lemma
Let $T$ be the set of theorems of some theory in the language of arithmetic which contains minimal arithmetic. For any formula $\map B y$ in the language of arithmetic, there is a sentence $G$ such that :$T \vdash G \leftrightarrow \map B {\hat G}$ where $\hat G$ is the Gödel number of $G$ (more accurately, it is the term in the language of arithmetic obtained by applying the function symbol $s$ to $0$ this many times). {{Disambiguate|Definition:Logical Formula}}
Tarski's Undefinability Theorem
Let $\ZZ$ be the standard structure $\struct {\Z, +, \cdot, s, <, 0}$ for the language of arithmetic. Let $\operatorname {Th}_\ZZ$ be the sentences which are true in $\ZZ$. Let $\Theta$ be the set of Gödel numbers of those sentences in $\operatorname {Th}_\ZZ$. $\Theta$ is not definable in $\operatorname {Th}_\ZZ$.
Equivalence of Definitions of Limit Point
{{TFAE|def = Limit Point of Set|view = limit point}} That is, let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. Let $H^{\complement}$ denote the relative complement of $H$ in $S$. Then the following conditions are equivalent for any point $x \in S$: :$(1): \quad$ Every open neighborhood $U$ of $x$ satisfies $H \cap \paren {U \setminus \set x} \ne \O$. :$(2): \quad x$ belongs to the closure of $H$ but is not an isolated point of $H$. :$(3): \quad x$ is an adherent point of $H$ but is not an isolated point of $H$. :$(4): \quad H^{\complement} \cup \set x$ is ''not'' a neighborhood of $x$.
Denseness Preserved in Coarser Topology
Let $T = \struct {S, \tau}$ be a topological space. Let $T' = \struct {S, \tau'}$ be another topological space on $S$ such that $\tau'$ is coarser than $\tau$. Let $H \subseteq S$ be everywhere dense in $T$. Then $H$ is also everywhere dense in $T'$.
Rationals are Dense in Compact Complement Topology
Let $T = \struct {\R, \tau^*}$ be the compact complement topology on $\R$. Let $\Q$ be the set of rational numbers. Then $\Q$ is everywhere dense in $T$.
Compact Complement Topology is Separable
Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$. Then $T$ is a separable space.
Either-Or Topology is Locally Connected
Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is a locally connected space.
Either-Or Topology is not Locally Arc-Connected
Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is not a locally arc-connected space.
Fort Space is Scattered
Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is a scattered space.
Dense-in-itself Subset of T1 Space is Infinite
Let $T = \struct {S, \tau_p}$ be a topological space which is $T_1$ (Fréchet). Let $H \subseteq T$ be dense-in-itself. Then $H$ is infinite.
Equal Chords in Circle
In a circle, equal chords are equally distant from the center, and chords that are equally distant from the center are equal in length. {{:Euclid:Proposition/III/14}}
Relative Lengths of Chords of Circles
Of chords in a circle, the diameter is the greatest, and of the rest the nearer to the center is always greater than the more remote. {{:Euclid:Proposition/III/15}}
Construction of Tangent from Point to Circle
From a given point outside a given circle, it is possible to draw a tangent to that circle. {{:Euclid:Proposition/III/17}}
Construction of Circle from Segment
{{:Euclid:Proposition/III/25}}
Equal Angles in Equal Circles
In equal circles, equal angles stand on equal arcs, whether at the center or at the circumference of those circles. {{:Euclid:Proposition/III/26}}
Angles on Equal Arcs are Equal
In equal circles, angles standing on equal arcs are equal to one another, whether at the center or at the circumference of those circles. {{:Euclid:Proposition/III/27}}
Bisection of Arc
It is possible to bisect an arc of a circle. {{:Euclid:Proposition/III/30}}
Relative Sizes of Angles in Segments
In a circle: : the angle in a semicircle is right : the angle in a segment greater than a semicircle is acute : the angle in a segment less than a semicircle is obtuse. Further: : the angle of a segment greater than a semicircle is obtuse : the angle of a segment less than a semicircle is acute. {{:Euclid:Proposition/III/31}}
Angles made by Chord with Tangent
Let $EF$ be a tangent to a circle $ABCD$, touching it at $B$. Let $BD$ be a chord of $ABCD$. Then: : the angle in segment $BCD$ equals $\angle DBE$ and:  : the angle in segment $BAD$ equals $\angle DBF$. {{:Euclid:Proposition/III/32}}
Construction of Segment on Given Line Admitting Given Angle
On any given line segment, it is possible to describe a segment of a circle which admits an angle equal to any given rectilineal angle. {{:Euclid:Proposition/III/33}}
Tangent Secant Theorem
Let $D$ be a point outside a circle $ABC$. Let $DB$ be tangent to the circle $ABC$. Let $DA$ be a straight line which cuts the circle $ABC$ at $A$ and $C$. Then $DB^2 = AD \cdot DC$. {{:Euclid:Proposition/III/36}}
Identity Element of Natural Number Multiplication is One
Let $1$ be the element one of $\N$. Then $1$ is the identity element of multiplication: :$\forall n \in \N: n \times 1 = n = 1 \times n$
Secant Secant Theorem
Let $C$ be a point external to a circle $ABED$. Let $CA$ and $CB$ be straight lines which cut the circle at $D$ and $E$ respectively. Then: : $CA \cdot CD = CB \cdot CE$
Converse of Tangent Secant Theorem
Let $D$ be a point outside a circle $ABC$. Let $DA$ be a straight line which cuts the circle $ABC$ at $A$ and $C$. Let $DB$ intersect the circle at $B$ such that $DB^2 = AD \cdot DC$. Then $DB$ is tangent to the circle $ABC$. {{:Euclid:Proposition/III/37}}
Fitting Chord Into Circle
Into a given circle, it is possible to fit a chord equal to a given line segment which is not greater than the diameter of the circle. {{:Euclid:Proposition/IV/1}}
Construction of Isosceles Triangle whose Base Angle is Twice Apex
It is possible to construct an isosceles triangle such that each of the angles at the base is twice that at the apex. {{:Euclid:Proposition/IV/10}}
Inscribing Regular Pentagon in Circle
In a given circle, it is possible to inscribe a regular pentagon. {{:Euclid:Proposition/IV/11}}
Inscribing Circle in Regular Pentagon
In any given regular pentagon it is possible to inscribe a circle. {{:Euclid:Proposition/IV/13}}
Inscribing Regular Hexagon in Circle
In a given circle, it is possible to inscribe a regular hexagon. {{:Euclid:Proposition/IV/15}}
Multiplication of Numbers is Left Distributive over Addition
{{:Euclid:Proposition/V/1}} That is, if $ma, mb, mc$ etc. be any equimultiples of $a, b, c$ etc., then: :$m a + m b + m c + \cdots = m \paren {a + b + c + \cdots }$
Multiplication of Numbers is Right Distributive over Addition
{{:Euclid:Proposition/V/2}} That is: :$ma + na + pa + \cdots = \paren {m + n + p + \cdots} a$
Multiples of Terms in Equal Ratios
Let $a, b, c, d$ be quantities. Let $a : b = c : d$ where $a : b$ denotes the ratio between $a$ and $b$. Then for any numbers $m$ and $n$: :$m a : n b = m c : n d$ {{:Euclid:Proposition/V/4}}
Ratios of Equal Magnitudes
{{:Euclid:Proposition/V/7}} That is: :$a = b \implies a : c = b : c$ :$a = b \implies c : a = c : b$
Relative Sizes of Ratios on Unequal Magnitudes
{{:Euclid:Proposition/V/8}} That is: :$a > b \implies a : c > b : c$ :$a > b \implies c : a < c : b$
Magnitudes with Same Ratios are Equal
{{:Euclid:Proposition/V/9}} That is: :$a : c = b : c \implies a = b$ :$c : a = c : b \implies a = b$
Relative Sizes of Magnitudes on Unequal Ratios
{{:Euclid:Proposition/V/10}} That is: :$a : c > b : c \implies a > b$ :$c : b > c : a \implies b < a$
Equality of Ratios is Transitive
{{:Euclid:Proposition/V/11}} That is: :$A : B = C : D, C : D = E : F \implies A : B = E : F$
Sum of Components of Equal Ratios
{{:Euclid:Proposition/V/12}} That is: :$a_1 : b_1 = a_2 : b_2 = a_3 : b_3 = \cdots \implies \left({a_1 + a_2 + a_3 + \cdots}\right) : \left({b_1 + b_2 + b_3 + \cdots}\right)$
Relative Sizes of Proportional Magnitudes
{{:Euclid:Proposition/V/13}} That is: :$a : b = c : d, c : d > e : f \implies a : b > e : f$
Relative Sizes of Components of Ratios
{{:Euclid:Proposition/V/14}} That is, if $a : b = c : d$ then: :$a > c \implies b > d$ :$a = c \implies b = d$ :$a < c \implies b < d$
Ratio Equals its Multiples
{{:Euclid:Proposition/V/15}} That is: :$a : b \implies ma = mb$
Proportional Magnitudes are Proportional Alternately
{{:Euclid:Proposition/V/16}} That is: :$a : b = c : d \implies a : c = b : d$
Magnitudes Proportional Compounded are Proportional Separated
{{:Euclid:Proposition/V/17}} That is: :$a : b = c : d \implies \left({a - b}\right) : b = \left({c - d}\right) : d$
Magnitudes Proportional Separated are Proportional Compounded
{{:Euclid:Proposition/V/18}} That is: :$a : b = c : d \implies \left({a + b}\right) : b = \left({c + d}\right) : d$
Proportional Magnitudes have Proportional Remainders
{{:Euclid:Proposition/V/19}} That is: :$a : b = c : d \implies \left({a - c}\right) : \left({b - d}\right) = a : b$ where $a : b$ denotes the ratio of $a$ to $b$.
Relative Sizes of Successive Ratios
{{:Euclid:Proposition/V/20}} That is, let: :$a : b = d : e$ :$b : c = e : f$ Then: :$a > c \implies d > f$ :$a = c \implies d = f$ :$a < c \implies d < f$
Relative Sizes of Elements in Perturbed Proportion
{{:Euclid:Proposition/V/21}} That is, let: :$a : b = e : f$ :$b : c = d : e$ Then: :$a > c \implies d > f$ :$a = c \implies d = f$ :$a < c \implies d < f$
Equality of Ratios Ex Aequali
{{:Euclid:Proposition/V/22}} That is, if: :$a : b = d : e$ :$b : c = e : f$ then: :$a : c = d : f$
Equality of Ratios in Perturbed Proportion
{{:Euclid:Proposition/V/23}} That is, if: :$a : b = e : f$ :$b : c = d : e$ then: :$a : c = d : f$
Sum of Antecedent and Consequent of Proportion
{{:Euclid:Proposition/V/25}} That is, if $a : b = c : d$ and $a$ is the greatest and $d$ is the least, then: :$a + d > b + c$
Areas of Triangles and Parallelograms Proportional to Base
{{:Euclid:Proposition/VI/1}} Let $ABC$ and $ACD$ be triangles. Let $EC, CF$ be parallelograms under the same height. Then: :$AC : CD = \triangle ABC : \triangle ACD = \Box EC : \Box CF$ where: :$AC : CD$ denotes the ratio of the length of $AC$ to that of $CD$ :$\triangle ABC : \triangle ACD$ denotes the ratio of the area of $\triangle ABC$ to that of $\triangle ACD$ :$\Box EC : \Box CF$ denotes the ratio of the area of parallelogram $EC$ to that of parallelogram $CF$.
Angle Bisector Theorem
Let $\triangle ABC$ be a triangle. Let $D$ lie on the base $BC$ of $\triangle ABC$. Then the following are equivalent: :$(1): \quad AD$ is the angle bisector of $\angle BAC$ :$(2): \quad BD : DC = AB : AC$ where $BD : DC$ denotes the ratio between the lengths $BD$ and $DC$. {{:Euclid:Proposition/VI/3}}
Equiangular Triangles are Similar
Let two triangles have the same corresponding angles. Then their corresponding sides are proportional. Thus, by definition, such triangles are similar. {{:Euclid:Proposition/VI/4}}
Triangles with One Equal Angle and Two Sides Proportional are Similar
Let two triangles have two corresponding sides which are proportional. Let the angles adjacent to both of these sides be equal. Then all of their corresponding angles are equal. Thus, by definition, such triangles are similar. {{:Euclid:Proposition/VI/6}}
Triangles with One Equal Angle and Two Other Sides Proportional are Similar
Let two triangles be such that one of the angles of one triangle equals one of the angles of the other. Let two corresponding sides which are adjacent to one of the other angles, be proportional. Let the third angle in both triangles be either both acute or both not acute. Then all of the corresponding angles of these triangles are equal. Thus, by definition, such triangles are similar. {{:Euclid:Proposition/VI/7}}
Zeckendorf's Theorem
Every positive integer has a unique Zeckendorf representation. That is: Let $n$ be a positive integer. Then there exists a unique increasing sequence of integers $\left\langle{c_i}\right\rangle$ such that: :$\forall i \in \N: c_i \ge 2$ :$c_{i + 1} > c_i + 1$ :$\displaystyle n = \sum_{i \mathop = 0}^k F_{c_i}$ where $F_m$ is the $m$th Fibonacci number. For any given $n$, such a $\left\langle{c_i}\right\rangle$ is unique.
Sum of Non-Consecutive Fibonacci Numbers
Let $S$ be a non-empty set of distinct non-consecutive Fibonacci numbers not containing $F_0$ or $F_1$. Let the largest element of $S$ be $F_j$. Then: :$\displaystyle \sum_{F_i \mathop \in S} F_i < F_{j + 1}$ That is, the sum of all the elements of $S$ is strictly less than the next largest Fibonacci number. That is, given some increasing sequence $\left\langle {c_i}\right\rangle$ satisfying $c_i \ge 2$ and $c_{i + 1} \ge c_i + 1$: :$\displaystyle F_{c_k + 1} > \sum_{i \mathop = 0}^k F_{c_i}$
Construction of Third Proportional Straight Line
Given any two straight lines of given length $a$ and $b$, it is possible to construct a third straight line of length $c$ such that $a : b = b : c$. {{:Euclid:Proposition/VI/11}}
Construction of Fourth Proportional Straight Line
Given three straight lines of lengths $a, b, c$, it is possible to construct a fourth straight line of length $d$ such that $a : b = c : d$. {{:Euclid:Proposition/VI/12}}
Construction of Mean Proportional
Given any two straight lines of length $a$ and $b$ it is possible to find a straight line of length $c$ such that $a : c = c : b$. {{:Euclid:Proposition/VI/13}}
Sides of Equal and Equiangular Parallelograms are Reciprocally Proportional
{{:Euclid:Proposition/VI/14}} Note: in the above, ''equal'' is to be taken to mean ''of equal area''.
Sides of Equiangular Triangles are Reciprocally Proportional
{{:Euclid:Proposition/VI/15}} Note: in the above, ''equal'' is to be taken to mean ''of equal area''.
Rectangles Contained by Proportional Straight Lines
{{:Euclid:Proposition/VI/16}} Note: in the above, ''equal'' is to be taken to mean ''of equal area''.
Construction of Similar Polygon
On any given straight line it is possible to construct a polygon similar to any given polygon. {{:Euclid:Proposition/VI/18}}
Triangles with Two Equal Angles are Similar
Two triangles which have two corresponding angles which are equal are similar.
Ratio of Areas of Similar Triangles
{{:Euclid:Proposition/VI/19}} That is, the ratio of the areas of the similar triangles is the square of the ratio of the corresponding sides.
Similarity of Polygons is Equivalence Relation
Let $A, B, C$ be polygons. If $A$ and $B$ are both similar to $C$, then $A$ is similar to $B$. {{:Euclid:Proposition/VI/21}} It is also worth noting that: :$A$ is similar to $A$, and so similarity between polygons is reflexive. :If $A$ is similar to $B$, then $B$ is similar to $A$, and so similarity between polygons is symmetric. Hence the relation of similarity between polygons is an equivalence relation.
Integral Multiple of an Algebraic Number
Let $K$ be a number field and $\alpha \in K$. Then there exists a positive $n \in \Z$ such that $n \alpha \in \mathcal O_K$. In this context, $ \mathcal O_K$ denotes the algebraic integers in $K$.
Schönemann-Eisenstein Theorem
Let $\map f x = a_d x^d + a_{d - 1} x^{d - 1} + \dotsb + a_0 \in \Z \sqbrk x$ be a polynomial over the ring of integers $\Z$. Let $p$ be a prime such that: :$(1): \quad p \divides a_i \iff i \ne d$ :$(2): \quad p^2 \nmid a_0$ where $p \divides a_i$ signifies that $p$ is a divisor of $a_i$. Then $f$ is irreducible in $\Q \sqbrk x$.
Isolated Point of Closure of Subset is Isolated Point of Subset
Let $\struct {T, \tau}$ be a topological space. Let $H \subseteq T$ be a subset of $T$. Let $\map \cl H$ denote the closure of $H$. Let $x \in \map \cl H$ be an isolated point of $\map \cl H$. Then $x$ is also an isolated point of $H$.
Disjoint Open Sets remain Disjoint with one Closure
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A, B \in \tau$ such that $A \cap B = \varnothing$. Then: : $A^- \cap B = \varnothing$ where $A^-$ denotes the closure of $A$.