hcju/proofwikips huggingface dataset:
Primitive of Power of x by Inverse Hyperbolic Sine of x over a
:$\displaystyle \int x^m \sinh^{-1} \frac x a \ \mathrm d x = \frac {x^{m + 1} } {m + 1} \sinh^{-1} \frac x a- \frac 1 {m + 1}  \int \frac {x^{m + 1} } {\sqrt {x^2 + a^2} } \ \mathrm d x + C$
Primitive of Power of x by Inverse Hyperbolic Cotangent of x over a
:$\displaystyle \int x^m \coth^{-1} \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \coth^{-1} \frac x a - \frac a {m + 1} \int \frac {x^{m + 1} } {a^2 - x^2} \rd x + C$
Primitive of Power of x by Inverse Hyperbolic Cosecant of x over a
:$\displaystyle \int x^m \operatorname{csch}^{-1} \frac x a \rd x = \begin{cases} \displaystyle \frac {x^{m + 1} } {m + 1} \operatorname{csch}^{-1} \frac x a + \frac 1 {m + 1} \int \frac {x^m} {\sqrt {x^2 + a^2} } \rd x + C & : x > 0 \\ \displaystyle \frac {x^{m + 1} } {m + 1} \operatorname{csch}^{-1} \frac x a - \frac 1 {m + 1} \int \frac {x^m} {\sqrt {x^2 + a^2} } \rd x + C & : x < 0 \\ \end{cases}$
Hyperbolic Tangent Half-Angle Substitution
:$\displaystyle \int \map F {\sinh x, \cosh x} \rd x = 2 \int \map F {\frac {2 u} {1 - u^2}, \frac {1 + u^2} {1 - u^2} } \frac {\d u} {1 - u^2}$ where $u = \tanh \dfrac x 2$.
Reciprocal of One Plus Cosine
:$\dfrac 1 {1 + \cos x} = \dfrac 1 2 \sec^2 \dfrac x 2$
Reciprocal of One Minus Cosine
:$\dfrac 1 {1 - \cos x} = \dfrac 1 2 \csc^2 \left({\dfrac x 2}\right)$
Irrationality of Logarithm
Let $a, b \in \N_{>0}$ such that both $\nexists m, n \in \N_{>0}: a^m = b^n$. Then $\log_b a$ is irrational.
Tangent of 22.5 Degrees
:$\tan 22.5^\circ = \tan \dfrac \pi 8 = \sqrt 2 - 1$
Tangent of 67.5 Degrees
:$\tan 67.5 \degrees = \tan \dfrac {3 \pi} 8 = \sqrt 2 + 1$
Sine of x plus Cosine of x/Cosine Form
:$\sin x + \cos x = \sqrt 2 \, \map \cos {x - \dfrac \pi 4}$
Sine of x minus Cosine of x/Sine Form
:$\sin x - \cos x = \sqrt 2 \sin \left({x - \dfrac \pi 4}\right)$
Sine of x minus Cosine of x
=== Sine Form === {{:Sine of x minus Cosine of x/Sine Form}} === Cosine Form === {{:Sine of x minus Cosine of x/Cosine Form}}
Sine of x minus Cosine of x/Cosine Form
:$\sin x - \cos x = \sqrt 2 \, \map \cos {x - \dfrac {3 \pi} 4}$
Multiple of Sine plus Multiple of Cosine
=== Cosine Form === {{:Multiple of Sine plus Multiple of Cosine/Cosine Form}} === Sine Form === {{:Multiple of Sine plus Multiple of Cosine/Sine Form}}
Sum Formulas for Sine and Cosine
=== Sine of x plus Cosine of x: Sine Form === {{:Sine of x plus Cosine of x/Sine Form}} === Sine of x plus Cosine of x: Cosine Form === {{:Sine of x plus Cosine of x/Cosine Form}} === Sine of x minus Cosine of x: Sine Form === {{:Sine of x minus Cosine of x/Sine Form}} === Sine of x minus Cosine of x: Cosine Form === {{:Sine of x minus Cosine of x/Cosine Form}} === Cosine of x minus Sine of x: Sine Form === {{:Cosine of x minus Sine of x/Sine Form}} === Cosine of x minus Sine of x: Cosine Form === {{:Cosine of x minus Sine of x/Cosine Form}} == Multiple of Sine plus Multiple of Cosine == {{:Multiple of Sine plus Multiple of Cosine}}
Multiple of Sine plus Multiple of Cosine/Cosine Form
:$p \sin x + q \cos x = \sqrt {p^2 + q^2} \map \cos {x + \arctan \dfrac {-p} q}$
Multiple of Sine plus Multiple of Cosine/Sine Form
:$p \sin x + q \cos x = \sqrt {p^2 + q^2} \sin \left({x + \arctan \dfrac q p}\right)$
Primitive of Cosine of a x over Sine of a x plus phi
:$\displaystyle \int \frac {\cos a x \rd x} {\map \sin {a x + \phi} } = \frac {\ln \size {\map \sin {a x + \phi} } } {a \cos \phi} + \tan \phi \int \frac {\sin a x \rd x} {\map \sin {a x + \phi} } + C$
Primitive of Sine of a x over Sine of a x plus phi
:$\displaystyle \int \frac {\sin a x \ \mathrm d x} {\sin \left({a x + \phi}\right)} = \frac x {\cos \phi} - \tan \phi \int \frac {\cos a x \ \mathrm d x} {\sin \left({a x + \phi}\right)} + C$
Reciprocal of One Plus Hyperbolic Cosine
:$\dfrac 1 {1 + \cosh x} = \dfrac 1 2 \operatorname{sech}^2 \dfrac x 2$
Reciprocal of Hyperbolic Cosine Minus One
:$\dfrac 1 {\cosh x - 1} = \dfrac 1 2 \csch^2 \dfrac x 2$
Triangle Right-Angle-Hypotenuse-Side Equality
If two triangles have: : one right angle each : the sides opposite to the right angle equal : another two respective sides equal they will also have: : their third sides equal : the remaining two angles equal to their respective remaining angles.
Internal Angles of Regular Polygon
The size $A$ of each internal angle of a regular $n$-gon is given by: :$A = \dfrac {\paren {n - 2} 180 \degrees} n$
Sum of Internal Angles of Polygon
The sum $S$ of all internal angles of a polygon with $n$ sides is given by the formula $S = \paren {n - 2} 180 \degrees$.
Five Platonic Solids
There exist exactly five platonic solids: :$\paren 1: \quad$ the regular tetrahedron :$\paren 2: \quad$ the cube :$\paren 3: \quad$ the regular octahedron :$\paren 4: \quad$ the regular dodecahedron :$\paren 5: \quad$ the regular icosahedron. {{:Euclid:Proposition/XIII/18/Endnote}}
Geometric Sequence with Coprime Extremes is in Lowest Terms
Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be a geometric sequence of integers. Let: :$a_0 \perp a_n$ where $\perp$ denotes coprimality. Then $G_n$ is in its lowest terms.
Measurements of Common Angles/Straight Angle
The measurement of a straight angle is $\dfrac{360^\circ} 2 = 180^\circ$ or $\dfrac {2 \pi} 2 = \pi$.
Measurements of Common Angles/Right Angle
The measurement of a right angle is $\dfrac {180 \degrees} 2 = 90 \degrees$ or $\dfrac \pi 2$.
Measurements of Common Angles/Full Angle
A full angle is equal to one full rotation.
Measurements of Common Angles/Acute Angle
An acute angle measures $\theta$, where: :$0 \degrees < \theta < 90 \degrees$ or: :$0 < \theta < \dfrac \pi 2$
Measurements of Common Angles/Obtuse Angle
An obtuse angle measures $\theta$, where: : $90^\circ < \theta < 180^\circ$ or: : $\dfrac \pi 2 < \theta < \pi$
Measurements of Common Angles/Reflex Angle
A reflex angle measures $\theta$, where: : $180^\circ < \theta < 360^\circ$ or: : $\pi < \theta < 2 \pi$
Supplementary Interior Angles implies Parallel Lines
Given two infinite straight lines which are cut by a transversal, if the interior angles on the same side of the transversal are supplementary, then the lines are parallel.
Equal Corresponding Angles implies Parallel Lines
Given two infinite straight lines which are cut by a transversal, if the corresponding angles are equal, then the lines are parallel.
Equivalence of Definitions of Tangent Vector
Let $M$ be a smooth manifold. Let $m \in M$ be a point.  Let $V$ be an open neighborhood of $m$.  Let $C^\infty \left({V, \R}\right)$ be defined as the set of all smooth mappings $f: V \to \R$. {{TFAE|def = Tangent Vector}}
Parallelism implies Equal Alternate Angles
Given two infinite straight lines which are cut by a transversal, if the lines are parallel, then the alternate angles are equal.
Parallelism implies Equal Corresponding Angles
Given two infinite straight lines which are cut by a transversal, if the lines are parallel, then the corresponding angles are equal.
Proportion is Reflexive
Proportion is a reflexive relation. That is, every real variable is proportional to itself: :$\forall x \in \R: x \propto x$
Proportion is Symmetric
Proportion is a symmetric relation. That is: :$\forall x, y \in \R: x \propto y \implies y \propto x$
Proportion is Transitive
Proportion is a transitive relation. That is: :$\forall x,y,z \in \R: x \propto y \land y \propto z \implies x \propto z$
Proportion is Equivalence Relation
Proportion is an equivalence relation.
Parallelism implies Supplementary Interior Angles
Given two infinite straight lines which are cut by a transversal, if the lines are parallel, then the interior angles on the same side of the transversal are supplementary.
Sine of 18 Degrees
:$\sin 18 \degrees = \sin \dfrac \pi {10} = \dfrac {\sqrt 5 - 1} 4$ where $\sin$ denotes the sine function.
Cosine of 72 Degrees
:$\cos 72 \degrees = \cos \dfrac {2 \pi} 5 = \dfrac {\sqrt 5 - 1} 4$
Sine of 72 Degrees
:$\sin 72 \degrees = \sin \dfrac {2 \pi} 5 = \dfrac {\sqrt{10 + 2 \sqrt 5} } 4$ where $\sin$ denotes the sine function.
Cosine of 18 Degrees
:$\cos 18^\circ = \cos \dfrac \pi {10} = \dfrac {\sqrt{10 + 2 \sqrt 5}} 4$ where $\cos$ denotes the cosine function.
Sine of 3 Degrees
:$\sin 3^\circ = \sin \dfrac \pi {60} = \dfrac {\sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16}$ where $\sin$ denotes the sine function.
Cosine of 3 Degrees
:$\cos 3^\circ = \cos \dfrac {\pi} {60} = \dfrac{ \sqrt{30} - \sqrt{10} - \sqrt 6 + \sqrt 2 + 2 \sqrt{15 + 3 \sqrt 5} + 2 \sqrt{5 + \sqrt 5} } {16}$ where $\cos$ denotes the cosine function.
Square Root of Sum as Sum of Square Roots
Let $a, b \in \R, a \ge b$. Then: :$\sqrt {a + b} = \sqrt {\dfrac a 2 + \dfrac {\sqrt {a^2 - b^2}} 2} + \sqrt {\dfrac a 2 - \dfrac {\sqrt {a^2 - b^2}} 2}$
Sine of 1 Degree
:$\sin 1^\circ = \sin \dfrac {\pi} {180} = $ where $\sin$ denotes the sine function.
Rectangle is Parallelogram
Let $ABCD$ be a rectangle. Then $ABCD$ is a parallelogram.
Line at Right Angles to Diameter of Circle/Porism
{{:Euclid:Proposition/III/16/Porism}}
Characteristic Function of Union/Variant 2
:$\chi_{A \mathop \cup B} = \chi_A + \chi_B - \chi_{A \mathop \cap B}$
Characteristic Function of Intersection/Variant 1
:$\chi_{A \cap B} = \chi_A \chi_B$
Sum of Square Roots as Square Root of Sum
:$\sqrt a + \sqrt b = \sqrt {a + b + \sqrt {4 a b} }$
Divisor of One of Coprime Numbers is Coprime to Other
Let $a, b \in \N$ be numbers such that $a$ and $b$ are coprime: :$a \perp b$ Let $c > 1$ be a divisor of $a$: :$c \divides a$ Then $c$ and $b$ are coprime: :$c \perp b$ {{:Euclid:Proposition/VII/23}}
Integer Coprime to all Factors is Coprime to Whole/Proof 1
Let $a, b \in \Z$. Let $\displaystyle b = \prod_{j \mathop = 1}^r b_j$ Let $a$ be coprime to each of $b_1, \ldots, b_r$. Then $a$ is coprime to $b$.
Integer Coprime to all Factors is Coprime to Whole/Proof 2
Let $a, b, c \in \Z$. Let $c$ be coprime to each of $a$ and $b$. Then $c$ is coprime to $a b$. {{:Euclid:Proposition/VII/24}}
Square of Coprime Number is Coprime
Let $a$ and $b$ be coprime integers: :$a, b \in \Z: a \perp b$ Then: :$a^2 \perp b$ {{:Euclid:Proposition/VII/25}}
Product of Coprime Pairs is Coprime
Let $a, b, c, d$ be integers. Let:  :$a \perp c, b \perp c, a \perp d, b \perp d$ where $a \perp c$ denotes that $a$ and $c$ are coprime. Then: :$a b \perp c d$ {{:Euclid:Proposition/VII/26}}
Powers of Coprime Numbers are Coprime
Let $a, b$ be coprime integers: :$a \perp b$ Then: :$\forall n \in \N_{>0}: a^n \perp b^n$ {{:Euclid:Proposition/VII/27}}
Numbers are Coprime iff Sum is Coprime to Both
Let $a, b$ be integers. Then: :$a \perp b \iff a \perp \paren {a + b}$ where $a \perp b$ denotes that $a$ and $b$ are coprime. {{:Euclid:Proposition/VII/28}}
Natural Number is Prime or has Prime Factor
Let $a$ be a natural number greater than $1$. Then either: :$a$ is a prime number or: :there exists a prime number $p \ne a$ such that $p \divides a$ where $\divides$ denotes '''is a divisor of'''. {{:Euclid:Proposition/VII/32}}
Negative of Real Function that Increases Without Bound
Let $f: \R \to \R$ be a real function. Then: :$(1): \quad \displaystyle \lim_{x \mathop \to +\infty} \map f x = +\infty \implies \lim_{x \mathop \to +\infty} -\map f x = -\infty$ :$(2): \quad \displaystyle \lim_{x \mathop \to -\infty} \map f x = +\infty \implies \lim_{x \mathop \to -\infty} -\map f x = -\infty$
Integer Divided by Divisor is Integer
Let $a, b \in \N$. Then: :$b \divides a \implies \dfrac 1 b \times a \in \N$ where $\divides$ denotes divisibilty. {{:Euclid:Proposition/VII/37}}
Divisor is Reciprocal of Divisor of Integer
Let $a, b, c \in \Z_{>0}$. Then: :$b = \dfrac 1 c \times a \implies c \divides a$ where $\divides$ denotes divisibilty. {{:Euclid:Proposition/VII/38}}
Negative of Real Function that Decreases Without Bound
Let $f: \R \to \R$ be a real function. Then: :$(1): \quad \displaystyle \lim_{x \mathop \to +\infty} \map f x = -\infty \implies \lim_{x \mathop \to +\infty} -\map f x = +\infty$ :$(2): \quad \displaystyle \lim_{x \mathop \to -\infty} \map f x = -\infty \implies \lim_{x \mathop \to -\infty} -\map f x = +\infty$
Construction of Geometric Sequence in Lowest Terms
It is possible to find a geometric sequence of integers $G_n$ of length $n + 1$ with a given common ratio such that $G_n$ is in its lowest terms. {{:Euclid:Proposition/VIII/2}}
Geometric Sequence in Lowest Terms has Coprime Extremes
A geometric sequence of integers in lowest terms has extremes which are coprime. {{:Euclid:Proposition/VIII/3}}
First Element of Geometric Sequence not dividing Second
Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers of length $n$. Let $a_0$ not be a divisor of $a_1$. Then: :$\forall j, k \in \set {0, 1, \ldots, n}, j \ne k: a_j \nmid a_k$ That is, if the initial term of $P$ does not divide the second, no term of $P$ divides any other term of $P$. {{:Euclid:Proposition/VIII/6}}
First Element of Geometric Sequence that divides Last also divides Second
Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers of length $n$. Let $a_0$ be a divisor of $a_n$. Then $a_0$ is a divisor of $a_2$. {{:Euclid:Proposition/VIII/7}}
Geometric Sequences in Proportion have Same Number of Elements
Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers of length $n$. Let $r$ be the common ratio of $P$. Let $Q = \sequence {b_j}_{0 \mathop \le j \mathop \le m}$ be a geometric sequence of integers of length $m$. Let $r$ be the common ratio of $Q$. Let $b_0$ and $b_m$ be such that $\dfrac {b_0} {b_m} = \dfrac {a_0} {a_n}$. Then $m = n$. {{:Euclid:Proposition/VIII/8}}
Elements of Geometric Sequence between Coprime Numbers
Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers of length $n + 1$. Let $a_0$ be coprime to $a_n$. Then there exist geometric sequences of integers $Q_1$ and $Q_2$ of length $n + 1$ such that: :the initial term of both $Q_1$ and $Q_2$ is $1$ :the final term of $Q_1$ is $a_0$ :the final term of $Q_2$ is $a_n$. {{:Euclid:Proposition/VIII/9}}
Product of Geometric Sequences from One
Let $Q_1 = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ and $Q_2 = \sequence {b_j}_{0 \mathop \le j \mathop \le n}$ be geometric sequences of integers of length $n + 1$. Let $a_0 = b_0 = 1$. Then the sequence $P = \sequence {p_j}_{0 \mathop \le j \mathop \le n}$ defined as: :$\forall j \in \set {0, \ldots, n}: p_j = b^j a^{n - j}$ is a geometric sequence. {{:Euclid:Proposition/VIII/10}}
Powers of Elements of Geometric Sequence are in Geometric Sequence
Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers. Then the sequence $Q = \sequence {b_j}_{0 \mathop \le j \mathop \le n}$ defined as: :$\forall j \in \set {0, 1, \ldots, n}: b_j = a_j^m$ where $m \in \Z_{>0}$, is a geometric sequence. {{:Euclid:Proposition/VIII/13}}
Number divides Number iff Square divides Square
Let $a, b \in \Z$. Then: :$a^2 \divides b^2 \iff a \divides b$ where $\divides$ denotes integer divisibility. {{:Euclid:Proposition/VIII/14}}
Number divides Number iff Cube divides Cube
Let $a, b \in \Z$. Then: :$a^3 \divides b^3 \iff a \divides b$ where $\divides$ denotes integer divisibility. {{:Euclid:Proposition/VIII/15}}
Number does not divide Number iff Cube does not divide Cube
Let $a, b \in \Z$ be integers. Then: :$a \nmid b \iff a^3 \nmid b^3$ where $a \nmid b$ denotes that $a$ is not a divisor of $b$. {{:Euclid:Proposition/VIII/17}}
If First of Four Numbers in Geometric Sequence is Cube then Fourth is Cube
Let $P = \tuple {a, b, c, d}$ be a geometric sequence of integers. Let $a$ be a cube number. Then $d$ is also a cube number. {{:Euclid:Proposition/VIII/23}}
If Ratio of Square to Number is as between Two Squares then Number is Square
Let $a, b, c, d \in \Z$ be integers such that: :$\dfrac a b = \dfrac {c^2} {d^2}$ Let $a$ be a square number. Then $b$ is also a square number. {{:Euclid:Proposition/VIII/24}}
Square of Cube Number is Cube
Let $a \in \N$ be a natural number. Let $a$ be a cube number. Then $a^2$ is also a cube number. {{:Euclid:Proposition/IX/3}}
Cube Number multiplied by Cube Number is Cube
Let $a, b \in \N$ be natural numbers. Let $a$ and $b$ be cube numbers. Then $a b$ is also a cube number. {{:Euclid:Proposition/IX/4}}
Number Squared making Cube is itself Cube
Let $a \in \Z$ be an integer. Let $a^2$ be a cube number. Then $a$ is a cube number. {{:Euclid:Proposition/IX/6}}
Product of Composite Number with Number is Solid Number
Let $a, b \in \Z$ be positive integers. Let $a$ be a composite number. Then $a b$ is a solid number. {{:Euclid:Proposition/IX/7}}
Square Numbers are Similar Plane Numbers
Let $a$ and $b$ be square numbers. Then $a$ and $b$ are similar plane numbers.
Elements of Geometric Sequence from One where First Element is Power of Number
Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a geometric sequence of integers. Let $a_0 = 1$. Let $m \in \Z_{> 0}$. Let $a_1$ be the $m$th power of an integer. Then all the terms of $G_n$ are $m$th powers of  integers. {{:Euclid:Proposition/IX/9}}
Sum of Pair of Elements of Geometric Sequence with Three Elements in Lowest Terms is Coprime to other Element
Let $P = \tuple {a, b, c}$ be a geometric sequence of integers in its lowest terms. Then $\paren {a + b}$, $\paren {b + c}$ and $\paren {a + c}$ are all coprime to each of $a$, $b$ and $c$. {{:Euclid:Proposition/IX/15}}
Two Coprime Integers have no Third Integer Proportional
Let $a, b \in \Z_{>0}$ be integers such that $a$ and $b$ are coprime. Then there is no integer $c \in \Z$ such that: :$\dfrac a b = \dfrac b c$ {{:Euclid:Proposition/IX/16}}
Last Element of Geometric Sequence with Coprime Extremes has no Integer Proportional as First to Second
Let $G_n = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers such that $a_0 \ne 1$. Let $a_0 \perp a_n$, where $\perp$ denotes coprimality. Then there does not exist an integer $b$ such that: :$\dfrac {a_0} {a_1} = \dfrac {a_n} b$ {{:Euclid:Proposition/IX/17}}
Condition for Existence of Third Number Proportional to Two Numbers
Let $a, b, c \in \Z$ be integers. Let $\tuple {a, b, c}$ be a geometric sequence. In order for this to be possible, both of these conditions must be true: :$(1): \quad a$ and $b$ cannot be coprime :$(2): \quad a \divides b^2$ where $\divides$ denotes divisibility. {{:Euclid:Proposition/IX/18}}
Odd Divisor of Even Number also divides its Half
Let $a, b \in \Z$ be integers. Let $a$ be odd and $b$ be even. Let: :$a \divides b$ where $\divides$ denotes divisibility. Then: :$a \divides \dfrac b 2$  {{:Euclid:Proposition/IX/30}}
Odd Number Coprime to Number is also Coprime to its Double
Let $a, b \in \Z$ be integers. Let $a$ be odd. Let: : $a \perp b$ where $\perp$ denotes coprimality. Then: : $a \perp 2 b$ {{:Euclid:Proposition/IX/31}}
Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd
Let $a \in \Z$ be an integer such that: :$(1): \quad a$ is not a power of $2$ :$(2): \quad \dfrac a 2$ is an even integer. Then $a$ is both even-times even and even-times odd. {{:Euclid:Proposition/IX/34}}
Scaled Real Function that Increases Without Bound
Let $f: \R \to \R$ be a real function. Let $\lambda \in \R_{\ne 0}$ be a nonzero constant. Then: For $\lambda > 0$: :$\displaystyle \lim_{x \mathop \to +\infty} \map f x = +\infty \implies \lim_{x \mathop \to +\infty} \lambda \map f x = +\infty$ :$\displaystyle \lim_{x \mathop \to -\infty} \map f x = +\infty \implies \lim_{x \mathop \to -\infty} \lambda \map f x = + \infty$ For $\lambda < 0$: :$\displaystyle \lim_{x \mathop \to +\infty} \map f x = +\infty \implies \lim_{x \mathop \to +\infty} \lambda \map f x = -\infty$ :$\displaystyle \lim_{x \mathop \to -\infty} \map f x = +\infty \implies \lim_{x \mathop \to -\infty} \lambda \map f x = -\infty$
Scaled Real Function that Decreases Without Bound
Let $f: \R \to \R$ be a real function. Let $\lambda \in \R_{\ne 0}$ be a nonzero constant. Then: For $\lambda > 0$: :$\displaystyle \lim_{x \mathop \to +\infty} \map f x = -\infty \implies \lim_{x \mathop \to +\infty} \lambda \map f x = -\infty$ :$\displaystyle \lim_{x \mathop \to -\infty} \map f x = -\infty \implies \lim_{x \mathop \to -\infty} \lambda \map f x = -\infty$ For $\lambda < 0$: :$\displaystyle \lim_{x \mathop \to +\infty} \map f x = -\infty \implies \lim_{x \mathop \to +\infty} \lambda \map f x = +\infty$ :$\displaystyle \lim_{x \mathop \to -\infty} \map f x = -\infty \implies \lim_{x \mathop \to -\infty} \lambda \map f x = +\infty$
Existence of Fraction of Number Smaller than Given
Let $a, b \in \R_{>0}$ be two (strictly) positive real numbers, such that $a > b$. Let the sequence $\left\langle{a_n}\right\rangle$ be defined recursively as: :$a_i = \begin{cases} a & : i = 1 \\ a_{i - 1} - c: \dfrac {a_{i - 1} } 2 < c < a_{i - 1} & : i > 1 \end{cases}$ Then: : $\exists n \in \N_{>0}: a_n < b$ {{:Euclid:Proposition/X/1}}
Restriction of Continuous Mapping is Continuous/Topological Spaces
Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces. Let $M_1 \subseteq S_1$ be a subset of $S_1$. Let $f: S_1 \to S_2$ be a mapping which is continuous. Let $M_2 \subseteq S_2$ be a subset of $S_2$ such that $f\left[{M_1}\right] \subseteq M_2$. Let $f {\restriction_{M_1 \times M_2}}: M_1 \to M_2$ be the restriction of $f$ to $M_1 \times M_2$. Then $f {\restriction_{M_1 \times M_2}}$ is continuous, where $M_1$ and $M_2$ are equipped with the respective subspace topologies.
Restriction of Continuous Mapping is Continuous/Metric Spaces
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Let $S \subseteq M_1$ be a subset of $M_1$. Let $f: M_1 \to M_2$ be a mapping which is continuous at a point $\alpha \in S$. Let $f \restriction_S = g: S \to M_2$ be the restriction of $f$ to $S$. Then $g$ is continuous at $\alpha$.
Restriction of Inverse is Inverse of Restriction
Let $S_1$ and $S_2$ be sets. Let $f: S_1 \to S_2$ be a bijection. Let $S \subseteq S_1$ be a subset of $S_1$. Let $f^{-1}$ be the inverse of $f$. Let $f {\restriction_{S \times f \left[{S}\right]}}$ be the restriction of $f$ to $S \times f\left[{S}\right]$. Let $f^{-1} {\restriction_{f \left[{S}\right] \times S}}$ be the restriction of $f^{-1}$ to $f \left[{S}\right] \times S$. Then: : $f {\restriction_{S \times f \left[{S}\right]}}$ is a bijection and: : ${\left({f {\restriction_{S \times f \left[{S}\right]}} }\right)}^{-1} = f^{-1} {\restriction_{f \left[{S}\right] \times S}}$
Hölder's Inequality/General
Let $\struct {X, \Sigma, \mu}$ be a measure space. For $i = 1, \ldots, n$ let $p_i \in \R$ such that: :$\displaystyle \sum_{i \mathop = 1}^n \frac 1 {p_i} = 1$ Let $f_i \in \map {\LL^{p_i} } \mu, f_i: X \to \R$, where $\LL$ denotes Lebesgue space. Then their pointwise product $\displaystyle \prod_{i \mathop = 1}^n f_i$ is integrable, that is: :$\displaystyle \prod_{i \mathop = 1}^n f_i \in \map {\LL^1} \mu$ and: :$\displaystyle \norm {\prod_{i \mathop = 1}^n f_i}_1 = \int \size {\prod_{i \mathop = 1}^n f_i} \rd \mu \le \prod_{i \mathop = 1}^n \norm {f_i}_{p_i}$ where the various instances of $\norm {\, \cdot \,}$ signify $p$-seminorms.
Theorem of Even Perfect Numbers/Sufficient Condition
Let $n \in \N$ be such that $2^n - 1$ is prime. Then $2^{n - 1} \paren {2^n - 1}$ is perfect. {{:Euclid:Proposition/IX/36}}
Theorem of Even Perfect Numbers/Necessary Condition
Let $a \in \N$ be an even perfect number. Then $a$ is in the form: :$2^{n - 1} \paren {2^n - 1}$ where $2^n - 1$ is prime.
Condition for Commensurability of Roots of Quadratic Equation/Lemma
{{:Euclid:Proposition/X/17/Lemma}}
Condition for Incommensurability of Roots of Quadratic Equation
Consider the quadratic equation: :$(1): \quad a x - x^2 = \dfrac {b^2} 4$ Then $x$ and $a - x$ are incommensurable  $\sqrt {a^2 - b^2}$ and $a$ are incommensurable. {{:Euclid:Proposition/X/18}}
Product of Rationally Expressible Numbers is Rational/Lemma
{{:Euclid:Proposition/X/19/Lemma}}
Product of Rationally Expressible Numbers is Rational
Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be the lengths of rational line segments. Furthermore, let $\dfrac a b \in \Q$. Then, $a b \in \Q$. {{:Euclid:Proposition/X/19}}