hcju/proofwikips huggingface dataset:
Sign of Cosine
Let $x$ be a real number. Then: {{begin-eqn}} {{eqn | l = \cos x       | o = >       | r = 0       | c = if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ }} {{eqn | l = \cos x       | o = <       | r = 0       | c = if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ }} {{end-eqn}} where $\cos$ is the real cosine function.
Sign of Tangent
Let $x$ be a real number. Then: {{begin-eqn}} {{eqn | l = \tan x       | o = >       | r = 0       | c = if there exists an integer $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$ }} {{eqn | l = \tan x       | o = <       | r = 0       | c = if there exists an integer $n$ such that $\paren {n + \dfrac 1 2} \pi < x < \paren {n + 1} \pi$ }} {{end-eqn}} where $\tan$ denotes the tangent function.
Reciprocal of Strictly Negative Real Number is Strictly Negative
:$\forall x \in \R: x < 0 \implies \dfrac 1 x < 0$
Sign of Secant
Let $x$ be a real number. {{begin-eqn}} {{eqn | l = \sec x       | o = >       | r = 0       | c = if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ }} {{eqn | l = \sec x       | o = <       | r = 0       | c = if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ }} {{end-eqn}} where $\sec$ is the real secant function.
Cosine in terms of Tangent
{{begin-eqn}} {{eqn | l = \cos x       | r = +\frac 1 {\sqrt {1 + \tan^2 x} }       | c = if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ }} {{eqn | l = \cos x       | r = -\frac 1 {\sqrt {1 + \tan^2 x} }       | c = if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ }} {{end-eqn}}
Tangent in terms of Secant
{{begin-eqn}} {{eqn | l = \tan x       | r = +\sqrt {\sec^2 x - 1}       | c = if there exists an integer $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$ }} {{eqn | l = \tan x       | r = -\sqrt {\sec^2 x - 1}       | c = if there exists an integer $n$ such that $\paren {n + \dfrac 1 2} \pi < x < \paren {n + 1} \pi$ }} {{end-eqn}}
Union of Left-Total Relations is Left-Total
Let $S_1, S_2, T_1, T_2$ be sets or classes. Let $\mathcal R_1 \subseteq S_1 \times T_1$ and $\mathcal R_2 \subseteq S_2 \times T_2$ be left-total relations. Then $\mathcal R_1 \cup \mathcal R_2$ is left-total.
Union of Inverse is Inverse of Union
Let for $i \in \left\{1,2\right\}$ $\mathcal R_i \subseteq S_i \times T_i$ be a relation on $S_i \times T_i$. Let $\mathcal R_i^{-1} \subseteq T_i \times S_i$ be the inverse of $\mathcal R_i$. Then $\mathcal R_1^{-1} \cup \mathcal R_2^{-1} = \left(\mathcal R_1 \cup \mathcal R_2\right)^{-1}$
Condition for Darboux Integrability
Let $\closedint a b$ be a closed real interval. Let $f$ be a bounded real function defined on $\closedint a b$. Then $f$ is Darboux integrable : :for every $\epsilon \in \R_{>0}$, there exists a finite subdivision $S$ of $\closedint a b$ such that $\map U S – \map L S < \epsilon$ where :$\map U S$ is the upper sum of $f$ on $\closedint a b$ with respect to $S$ :$\map L S$ is the lower sum of $f$ on $\closedint a b$ with respect to $S$
Open Set minus Closed Set is Open
Let $T = \left({S, \tau}\right)$ be a topological space. For $A \subseteq S$ denote by $\complement_S \left({A}\right)$ the relative complement of $A$ in $S$. Let $U \in \tau$ and $\complement_S \left({V}\right) \in \tau$. Then: :$U \setminus V \in \tau$ and: :$\complement_S \left({V \setminus U}\right) \in \tau$
Union of Right-Total Relations is Right-Total
Let $S_1, S_2, T_1, T_2$ be sets or classes. Let $\mathcal R_1 \subseteq S_1 \times T_1$ and $\mathcal R_2 \subseteq S_2 \times T_2$ be right-total relations. Then $\mathcal R_1 \cup \mathcal R_2$ is right-total.
Primitive of x over a x + b squared by p x + q/Corollary
:$\displaystyle \int \frac {x \rd x} {\paren {a x + b}^2 \paren {p x + q} } = \frac 1 {b p - a q} \paren {\frac q {b p - a q} \ln \size {\frac {a x + b} {p x + q} } + \frac x {a x + b} } + C$
Arctangent of Imaginary Number
Let $x$ belong to the open real interval $\openint {-1} 1$. Then: :$\map {\tan^{-1} } {i x} = \dfrac i 2 \map \ln {\dfrac {1 + x} {1 - x} }$ where $\tan$ is the complex tangent function, $\ln$ is the real natural logarithm, and $i$ is the imaginary unit.
Equivalence of Definitions of Real Inverse Hyperbolic Tangent
Let $S$ denote the open real interval: : $S := \left({-1 \,.\,.\, 1}\right)$ {{TFAE|def = Real Inverse Hyperbolic Tangent}}
Square Root is Strictly Increasing
The positive square root function is strictly increasing, that is: :$ \forall x,y \in \R_{>0}: x < y \implies \sqrt x < \sqrt y$
Minimum of Real Hyperbolic Cosine Function
Let $x$ be a real number. Then: :$\cosh x \ge 1$ where $\cosh$ denotes the hyperbolic cosine function.
Exponential of Real Number is Strictly Positive
Let $x$ be a real number. Let $\exp$ denote the (real) exponential function. Then: :$\forall x \in \R : \exp x > 0$
Derivative of Power of Function
Let $\map u x$ be a differentiable real function of $x$. Let $n$ be a real number such that $n \ne -1$. Then: :$\map {\dfrac \d {\d x} } {\map u x^n} = n \map u x^{n - 1} \map {\dfrac \d {\d x} } {\map u x}$
Natural Logarithm of e is 1
:$\ln e = 1$
Real Inverse Hyperbolic Cosine is Strictly Increasing
The real inverse hyperbolic cosine function is strictly increasing, that is: :$\forall x, y \ge 1 : x < y \implies \cosh^{-1} x < \cosh^{-1} y$
Laplace Transform of Constant Multiple
Let $a \in \C$ or $\R$ be constant. Then: :$a \laptrans {\map f {a t} } = \map F {\dfrac s a}$
Laplace Transform of Function of t minus a
Let $g$ be the function defined as: :$\map g t = \begin{cases} \map f {t - a} & : t > a \\ 0 & : t \le a \end{cases}$ Then: :$\laptrans {\map g t} = e^{-a s} \map F s$
Seifert-van Kampen Theorem
The functor $\pi_1 : \mathbf{Top_\bullet} \to \mathbf{Grp}$ preserves pushouts of inclusions.
Functions of Independent Random Variables are Independent
Let $X$ and $Y$ be independent random variables on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $g$ and $h$ be real-valued functions defined on the codomains of $X$ and $Y$ respectively. Then $\map g X$ and $\map h Y$ are independent random variables.
Multiplication Property of Characteristic Functions
Let $X$ and $Y$ be independent random variables on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $\phi_X$ and $\phi_Y$ denote the characteristic functions of $X$ and $Y$ respectively. Then: :$\phi_{X + Y} = \phi_X \phi_Y$
Relationship between Limit Inferior and Lower Limit
Let $\struct {S, \tau}$ be a topological space. Let $f: S \to \R \cup \set {-\infty, \infty}$ be an extended real-valued function. Let $\sequence {s_n}_{n \mathop \in \N}$ be a convergent sequence in $S$ such that $s_n \to \bar s$. Then the lower limit of $f$ at $\bar s$ is bounded above by the limit inferior of $\sequence {\map f {s_n} }$: :$\displaystyle \liminf_{s \mathop \to \bar s} \map f s \le \liminf_{n \mathop \to \infty} \map f {s_n}$
Sequence on Finite Product Space Converges to Point iff Projections Converge to Projections of Point
Let $N \in \N$. For all $k \in \set {1, \ldots, N}$, let $T_k = \struct {X_k, \tau_k}$ be topological spaces. Let $\displaystyle X = \prod_{k \mathop = 1}^N X_k$ be the cartesian product of $X_1, \ldots, X_N$. Let $\tau$ be the product topology on $X$. Denote by $\pr_k : X \to X_k$ the projection from $X$ onto $X_k$. Let $\sequence {x_n}$ be a sequence on $X$ and let $x \in X$. Then $\sequence {x_n}$ converges to $x$ : :for all $k \in \set {1, \ldots, N}$ the sequence $\sequence {\map {\pr_k} {x_n} }$ converges to $\map {\pr_k} x$.
Real Convergent Sequence is Cauchy Sequence
Every convergent real sequence in $\R$ is a Cauchy sequence.
Real Sequence is Cauchy iff Convergent
Let $\sequence {a_n}$ be a sequence in $\R$. Then $\sequence {a_n}$ is a Cauchy sequence  it is convergent.
Complex Sequence is Cauchy iff Convergent
Let $\sequence {z_n}$ be a complex sequence. Then $\sequence {z_n}$ is a Cauchy sequence  it is convergent.
Sum of Arctangents
:$\arctan a + \arctan b = \arctan \dfrac {a + b} {1 - a b}$ where $\arctan$ denotes the arctangent.
Difference of Arctangents
:$\arctan a - \arctan b = \arctan \dfrac {a - b} {1 + a b}$ where $\arctan$ denotes the arctangent.
Sum of Arccotangents
:$\arccot a + \arccot b = \arccot \dfrac {a b - 1} {a + b}$ where $\arccot$ denotes the arccotangent.
Difference of Arccotangents
:$\arccot a - \arccot b = \arccot \dfrac {a b + 1} {a - b}$ where $\arccot$ denotes the arccotangent.
Multiple Angle Formula for Tangent
:$\displaystyle \tan \left({n \theta}\right) = \frac {\displaystyle \sum_{i \mathop = 0}^{\left\lfloor{\frac {n - 1} 2}\right\rfloor} \left({-1}\right)^i \binom n {2 i + 1} \tan^{2 i + 1}\theta} {\displaystyle \sum_{i \mathop = 0}^{\left\lfloor{\frac n 2}\right\rfloor} \left({-1}\right)^i \binom n {2 i} \tan^{2 i}\theta}$
Modulus of Complex Number equals its Distance from Origin
The modulus of a complex number equals its distance from the origin on the complex plane.
Empty Set is Countable
The empty set $\O$ is countable.
Area of Isosceles Triangle in terms of Sides
Let $\triangle ABC$ be an isosceles triangle whose apex is $A$. Let $r$ be the length of a leg of $\triangle ABC$. Let $b$ be the length of the base of $\triangle ABC$. Then the area $\mathcal A$ of $\triangle ABC$ is given by: :$\mathcal A = \dfrac b 4 \sqrt{4 r^2 - b^2}$
Even Function Times Even Function is Even
Let $X \subset \R$ be a symmetric set of real numbers: :$\forall x \in X: -x \in X$ Let $f, g: X \to \R$ be two even functions. Let $f \cdot g$ denote the pointwise product of $f$ and $g$. Then $\paren {f \cdot g}: X \to \R$ is also an even function.
Odd Function Times Even Function is Odd
Let $X \subset \R$ be a symmetric set of real numbers: :$\forall x \in X: -x \in X$ Let $f: X \to \R$ be an odd function. Let $g: X \to \R$ be an even function. Let $f \cdot g$ denote the pointwise product of $f$ and $g$. Then $\paren {f \cdot g}: X \to \R$ is an odd function.
Odd Function Times Odd Function is Even
Let $S \subset \R$ be a symmetric set of real numbers: :$\forall x \in S: -x \in X$ Let $f, g: X \to \R$ be two odd functions. Let $f \cdot g$ denote the pointwise product of $f$ and $g$. Then $\paren {f \cdot g}: S \to \R$ is an even function.
Mellin Transform of Exponential
Let $a$ be a complex constant and $e^t$ be the complex exponential. Let $\MM$ be the Mellin transform. Then: :$\map {\MM \set {e^{-a t} } } s = a^{-s} \, \map \Gamma s$ where $\map \Re a, \map \Re s > 0$
Mellin Transform of Dirac Delta Function
Let $\delta \left({t}\right)$ be the Dirac delta function. Let $c$ be a positive real number. Let $\mathcal M$ be the Mellin transform. Then: :$\mathcal M \left\{ { \delta_c \left({t}\right)} \right\} \left({s}\right) = c^{s-1}$
Mellin Transform of Power Times Function
Let $t^n: \R \to \R$ be $t$ to the $n$th power for some $n \in \N_{\ge 0}$. Let $\mathcal M$ be the Mellin transform. Then: :$\mathcal M \left\{ {t^n f \left({t}\right)} \right\} \left({s}\right) = \mathcal M \left\{ {f \left({t}\right)}\right\} \left({s + n}\right)$ given that both transforms exist.
Mellin Transform of Dirac Delta Function by Function
Let $f: \R \to \R$ be a function. Let $c \in \R_{>0}$ be a positive constant real number. Let $\delta_c \left({t}\right)$ be the Dirac delta function. Let $\mathcal M$ be the Mellin transform. Then: :$\mathcal M \left\{ {\delta_c \left({t}\right) f \left({t}\right)} \right\} \left({s}\right) = c^{s - 1} f \left({c}\right)$
Unity of Ring is Idempotent
Let $\left({R, +, \circ}\right)$ be a ring with unity whose unity is $1_R$. Then $1_R$ is an idempotent element of $R$ under the ring product $\circ$: :$1_R \circ 1_R = 1_R$
Mellin Transform of Higher Order Exponential
Let $a$ be a complex constant. Let $n$ be a natural number. Let $e^t$ be the complex exponential of $t$. Let $\MM$ be the Mellin transform. Then: :$\map {\MM \set {e^{-a t^n} } } s = \dfrac {a^{-s/n} } n \map \Gamma {\dfrac s n}$ where $\map \Gamma z$ is the Gamma function and $\map \Re a$, $\map \Re s > 0$.
Complex Numbers as Quotient Ring of Real Polynomial
Let $\C$ be the set of complex numbers. Let $P \left[{x}\right]$ be the set of polynomials over real numbers, where the coefficients of the polynomials are real. Let $\left\langle{x^2 + 1}\right\rangle = \left\{ {Q \left({x}\right) \left({x^2 + 1}\right): Q \left({x}\right) \in P \left[{x}\right]}\right\}$ be the ideal generated by $x^2 + 1$ in $P \left[{x}\right]$. Let $D = P \left[{x}\right] / \left\langle{x^2 + 1}\right\rangle$ be the quotient of $P \left[{x}\right]$ modulo $\left\langle{x^2 + 1}\right\rangle$. Then: :$\left({\C, +, \times}\right) \cong \left({D, +, \times}\right)$
Quaternion Modulus in Terms of Conjugate
Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion. Let $\left\vert{\mathbf x}\right\vert$ be the modulus of $\mathbf x$. Let $\overline{\mathbf x}$ be the conjugate of $\mathbf x$. Then: : $\left\vert{\mathbf x}\right\vert^2 \mathbf 1 = \mathbf x \overline{\mathbf x}$
Sufficient Condition for Quaternion Multiplication to Commute
In general, quaternion multiplication does not commute. But, for $\mathbf x,\mathbf y \in \H$, $\mathbf x \times \mathbf y = \mathbf y \times \mathbf x$ if any one of the following conditions hold: {{begin-eqn}} {{eqn | n = 1a       | l = \mathbf x, \mathbf y       | o = \in       | r = \set {a \mathbf 1 + b \mathbf i + 0 \mathbf j + 0 \mathbf k: a, b \in \R} }} {{eqn | n = 1b       | l = \mathbf x, \mathbf y       | o = \in       | r = \set {a \mathbf 1 + 0 \mathbf i + c \mathbf j + 0 \mathbf k: a, c \in \R} }} {{eqn | n = 1c       | l = \mathbf x, \mathbf y       | o = \in       | r = \set {a \mathbf 1 + 0 \mathbf i + 0 \mathbf j + d \mathbf k: a, d \in \R} }} {{eqn | n = 2a       | l = \mathbf x       | o = \in       | r = \set {a \mathbf 1 + 0 \mathbf i + 0 \mathbf j + 0 \mathbf k: a \in \R} }} {{eqn | n = 2b       | l = \mathbf y       | o = \in       | r = \set {a \mathbf 1 + 0 \mathbf i + 0 \mathbf j + 0 \mathbf k: a \in \R} }} {{eqn | n = 3       | l = \mathbf x       | r = \paren {a \mathbf 1 + 0 \mathbf i + 0 \mathbf j + 0 \mathbf k} \times \overline {\mathbf y}: a \in \R }} {{end-eqn}}
Complex Conjugation is Involution
Let $z = x + i y$ be a complex number. Let $\overline z$ denote the complex conjugate of $z$. Then the operation of complex conjugation is an involution: :$\overline {\paren {\overline z} } = z$
Quaternion Modulus of Conjugate
Let $z = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion. Let $\overline z$ be the conjugate of $z$. Let $\cmod z$ be the quaternion modulus of $z$. Then: :$\cmod {\overline z} = \cmod z$
Quaternion Conjugation is Involution
Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion. Let $\overline {\mathbf x}$ denote the quaternion conjugate of $\mathbf x$. Then the operation of quaternion conjugation is an involution: :$\overline {\paren {\overline {\mathbf x} } } = \mathbf x$
Sum of Quaternion Conjugates
Let $\mathbf x, \mathbf y \in \mathbb H$ be quaternions. Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$. Then: :$\overline {\mathbf x + \mathbf y} = \overline {\mathbf x} + \overline {\mathbf y}$
Principle of Recursive Definition for Peano Structure
Let $\struct {P, 0, s}$ be a Peano structure. Let $T$ be a set. Let $a \in T$. Let $g: T \to T$ be a mapping. Then there exists exactly one mapping $f: P \to T$ such that: :$\forall x \in P: \map f x = \begin{cases} a & : x = 0 \\ \map g {\map f n} & : x = \map s n \end{cases}$
Principle of Recursive Definition for Minimal Infinite Successor Set
Let $\omega$ be the minimal infinite successor set. Let $T$ be a set. Let $a \in T$. Let $g: T \to T$ be a mapping. Then there exists exactly one mapping $f: \omega \to T$ such that: :$\forall x \in \omega: \map f x = \begin {cases} a & : x = \O \\ \map g {\map f n} & : x = n^+ \end {cases}$ where $n^+$ is the successor set of $n$.
Ordering on 1-Based Natural Numbers is Trichotomy
Let $\N_{> 0}$ be the $1$-based natural numbers. Let $<$ be the strict ordering on $\N_{>0}$. Then exactly one of the following is true: :$(1): \quad a = b$ :$(2): \quad a > b$ :$(3): \quad a < b$ That is, $<$ is a trichotomy on $\N_{> 0}$.
Product of Quaternion Conjugates
Let $\mathbf x, \mathbf y \in \mathbb H$ be quaternions. Let $\overline{\mathbf x}$ be the conjugate of $\mathbf x$. Then: :$\overline{\mathbf x \times \mathbf y} = \overline{\mathbf y} \times \overline{\mathbf x}$ but in general: :$\overline{\mathbf x \times \mathbf y} \ne \overline{\mathbf x} \times \overline{\mathbf y}$
Quaternion Modulus of Product of Quaternions
Let $\mathbf x, \mathbf y$ be quaternions. Let $\left\vert{\mathbf x}\right\vert$ be the modulus of $\mathbf x$. Then: :$\left\vert{\mathbf x \mathbf y}\right\vert = \left\vert{\mathbf x}\right\vert \left\vert{\mathbf y}\right\vert$
Octonion Conjugation is Involution
Let $x = \tuple {a, b}: a, b \in \mathbb H$ be a octonion. Let $\overline x$ be the conjugate of $x$. Then: :$\overline \cdot: x \mapsto \overline x$ is an involution. That is: :$\overline {\paren {\overline x} } = x$
Ordering on 1-Based Natural Numbers is Compatible with Addition
Let $\N_{> 0}$ be the $1$-based natural numbers. Let $+$ denote addition on $\N_{>0}$. Let $<$ be the strict ordering on $\N_{>0}$. Then: :$\forall a, b, n \in \N_{>0}: a < b \implies a + n < b + n$ That is, $>$ is compatible with $+$ on $\N_{>0}$.
Ordering on 1-Based Natural Numbers is Compatible with Multiplication
Let $\N_{> 0}$ be the $1$-based natural numbers. Let $\times$ denote multiplication on $\N_{>0}$. Let $<$ be the strict ordering on $\N_{>0}$. Then: :$\forall a, b, n \in \N_{>0}: a < b \implies a \times n < b \times n$ That is, $>$ is compatible with $\times$ on $\N_{>0}$.
Integral between Limits is Independent of Direction
Let $f$ be a real function which is integrable on the interval $\openint a b$. Then: :$\ds \int_a^b \map f x \rd x = \int_a^b \map f {a + b - x} \rd x$
Countable Set equals Range of Sequence
Let $S$ be a set. Then $S$ is countable  there exists a sequence $\sequence {s_i}_{i \mathop \in N}$ where $N$ is a subset of $\N$ such that $S$ equals the range of $\sequence {s_i}_{i \mathop \in N}$.
Initial Segment of Natural Numbers determined by Zero is Empty
Let $\N_k$ denote the initial segment of the natural numbers determined by $k$: :$\N_k = \set {0, 1, 2, 3, \ldots, k - 1}$ Then $\N_0 = \O$.
Initial Segment of One-Based Natural Numbers determined by Zero is Empty
Let $\N^*_k$ denote the initial segment of the one-based natural numbers determined by $k$: :$\N^*_k = \set {1, 2, 3, \ldots, k - 1, k}$ Then $\N^*_0 = \O$.
Heine-Borel Theorem/Real Line/Closed and Bounded Interval
Let $\left[{a \,.\,.\, b}\right]$, $a < b$, be a closed and bounded real interval. Let $S$ be a set of open real sets. Let $S$ be a cover of $\left[{a \,.\,.\, b}\right]$. Then there is a finite subset of $S$ that covers $\left[{a \,.\,.\, b}\right]$.
Addition on 1-Based Natural Numbers is Cancellable
Let $\N_{> 0}$ be the $1$-based natural numbers. Let $+$ be addition on $\N_{>0}$. Then: :$\forall a, b, c \in \N_{>0}: a + c = b + c \implies a = b$ :$\forall a, b, c \in \N_{>0}: a + b = a + c \implies b = c$ That is, $+$ is cancellable on $\N_{>0}$.
Addition on 1-Based Natural Numbers is Cancellable for Ordering
Let $\N_{> 0}$ be the $1$-based natural numbers. Let $<$ be the strict ordering on $\N_{>0}$. Let $+$ be addition on $\N_{>0}$. Then: :$\forall a, b, c \in \N_{>0}: a + c < b + c \implies a < b$ :$\forall a, b, c \in \N_{>0}: a + b < a + c \implies b < c$ That is, $+$ is cancellable on $\N_{>0}$ for $<$.
Multiplication on 1-Based Natural Numbers is Cancellable
Let $\N_{> 0}$ be the $1$-based natural numbers. Let $\times$ be multiplication on $\N_{>0}$. Then: :$\forall a, b, c \in \N_{>0}: a \times c = b \times c \implies a = b$ :$\forall a, b, c \in \N_{>0}: a \times b = a \times c \implies b = c$ That is, $\times$ is cancellable on $\N_{>0}$.
Multiplication on 1-Based Natural Numbers is Cancellable for Ordering
Let $\N_{> 0}$ be the $1$-based natural numbers. Let $\times$ be multiplication on $\N_{>0}$. Let $<$ be the strict ordering on $\N_{>0}$. Then: :$\forall a, b, c \in \N_{>0}: a \times c < b \times c \implies a < b$ :$\forall a, b, c \in \N_{>0}: a \times b < a \times c \implies b < c$ That is, $\times$ is cancellable on $\N_{>0}$ for $<$.
Index of Trivial Subgroup is Cardinality of Group
Let $G$ be a group whose identity element is $e$. Let $\set e$ be the trivial subgroup of $G$. Then: :$\index G {\set e} = \order G$ where: :$\index G {\set e}$ denotes the index of $\set e$ in $G$ :$\order G$ denotes the cardinality of $G$.
Index is One iff Subgroup equals Group
Let $G$ be a group whose identity element is $e$. Let $H$ be a subgroup of $G$. Then: :$\index G H = 1 \iff G = H$ where $\index G H$ denotes the index of $H$ in $G$.
Set Equality is Equivalence Relation
Let $S$ be a set. Set equality is an equivalence relation on the power set $\powerset S$ of $S$.
Subset Relation is Antisymmetric
The relation "is a subset of" is antisymmetric: :$\paren {R \subseteq S} \land \paren {S \subseteq R} \iff R = S$
There Exists No Universal Set
There exists no set which is an absolutely universal set. That is: :$\neg \left({\exists \mathcal U: \forall T: T \in \mathcal U}\right)$ where $T$ is any arbitrary object at all. That is, a set that contains ''everything'' cannot exist.
Union of Disjoint Singletons is Doubleton
Let $\set a$ and $\set b$ be singletons such that $a \ne b$. Then: :$\set a \cup \set b = \set {a, b}$
Distributive Laws
== Intersection Distributes over Union == {{:Intersection Distributes over Union}}