hcju/proofwikips huggingface dataset:
Sign of Odd Power
Let $x \in \R$ be a real number. Let $n \in \Z$ be an odd integer. Then: :$x^n = 0 \iff x = 0$ :$x^n > 0 \iff x > 0$ :$x^n < 0 \iff x < 0$ That is, the sign of an odd power matches the number it is a power of.
Product of Absolute Values on Ordered Integral Domain
Let $\struct {D, +, \times, \le}$ be an ordered integral domain whose zero is denoted by $0_D$. For all $a \in D$, let $\size a$ denote the absolute value of $a$. Then: :$\size a \times \size b = \size {a \times b}$
Negative of Absolute Value
Let $x \in \R$ be a real number. Let $\size x$ denote the absolute value of $x$. Then: :$-\size x \le x \le \size x$
Order of Squares in Ordered Ring
Let $\struct {R, +, \circ, \le}$ be an ordered ring whose zero is $0_R$ and whose unity is $1_R$. Let $x, y \in \struct {R, +, \circ, \le}$ such that $0_R \le x, y$. Then: :$x \le y \implies x \circ x \le y \circ y$ When $R$ is one of the standard sets of numbers, that is $\Z, \Q, \R$, then this translates into: :If $x, y$ are positive then $x \le y \implies x^2 \le y^2$.
Continuum Property
Let $S \subset \R$ be a non-empty subset of the set of real numbers such that $S$ is bounded above. Then $S$ admits a supremum in $\R$. This is known as the '''least upper bound property''' of the real numbers. Similarly, let $S \subset \R$ be a non-empty subset of the set of real numbers such that $S$ is bounded below. Then $S$ admits an infimum in $\R$. This is sometimes called the '''greatest lower bound property''' of the real numbers. The two properties taken together are called the '''continuum property of $\R$'''. This can also be stated as: :The set $\R$ of real numbers is Dedekind complete.
Complex Multiplication is Closed
The set of complex numbers $\C$ is closed under multiplication: :$\forall z, w \in \C: z \times w \in \C$
Complex Multiplication is Associative
The operation of multiplication on the set of complex numbers $\C$ is associative: :$\forall z_1, z_2, z_3 \in \C: z_1 \paren {z_2 z_3} = \paren {z_1 z_2} z_3$
Arithmetic Mean is Never Less than Harmonic Mean
Let $x_1, x_2, \ldots, x_n \in \R_{> 0}$ be strictly positive real numbers. Let $A_n $ be the arithmetic mean of $x_1, x_2, \ldots, x_n$. Let $H_n$ be the harmonic mean of $x_1, x_2, \ldots, x_n$. Then $A_n \ge H_n$.
Multiple of Supremum
Let $S \subseteq \R: S \ne \varnothing$ be a non-empty subset of the set of real numbers $\R$. Let $S$ be bounded above. Let $z \in \R: z > 0$ be a positive real number. Then: :$\displaystyle \map {\sup_{x \mathop \in S} } {z x} = z \map {\sup_{x \mathop \in S} } x$
Cauchy's Mean Theorem
Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers which are all positive. Let $A_n$ be the arithmetic mean of $x_1, x_2, \ldots, x_n$. Let $G_n$ be the geometric mean of $x_1, x_2, \ldots, x_n$. Then: :$A_n \ge G_n$ with equality holding : :$\forall i, j \in \set {1, 2, \ldots, n}: x_i = x_j$ That is,  all terms are equal.
Distance on Real Numbers is Metric
Let $x, y \in \R$ be real numbers. Let $\map d {x, y}$ be the distance between $x$ and $y$: :$\map d {x, y} = \size {x - y}$ Then $\map d {x, y}$ is a metric on $\R$. Thus it follows that $\tuple {\R, d}$ is a metric space.
Combination Theorem for Sequences/Real
Let $\sequence {x_n}$ and $\sequence {y_n}$ be sequences in $\R$. Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent to the following limits: :$\displaystyle \lim_{n \mathop \to \infty} x_n = l$ :$\displaystyle \lim_{n \mathop \to \infty} y_n = m$ Let $\lambda, \mu \in \R$. Then the following results hold: === Sum Rule === {{:Combination Theorem for Sequences/Real/Sum Rule}} === Difference Rule === {{:Combination Theorem for Sequences/Real/Difference Rule}} === Multiple Rule === {{:Combination Theorem for Sequences/Real/Multiple Rule}} === Combined Sum Rule === {{:Combination Theorem for Sequences/Real/Combined Sum Rule}} === Product Rule === {{:Combination Theorem for Sequences/Real/Product Rule}} === Quotient Rule === {{:Combination Theorem for Sequences/Real/Quotient Rule}}
Convergent Sequence in Metric Space is Bounded
Let $M = \struct {A, d}$ be a metric space. Let $\sequence {x_n}$ be a sequence in $M$ which is convergent, and so $x_n \to l$ as $n \to \infty$. Then $\sequence {x_n}$ is bounded.
Convergent Sequence Minus Limit
Let $X$ be  one of the standard number fields $\Q, \R, \C$. Let $\left \langle {x_n} \right \rangle$ be a sequence in $X$ which converges to $l$. That is: : $\displaystyle \lim_{n \mathop \to \infty} x_n = l$ Then: : $\displaystyle \lim_{n \mathop \to \infty} \left|{x_n - l}\right| = 0$
Monotone Convergence Theorem (Real Analysis)
Every bounded monotone sequence is convergent.
One Plus Reciprocal to the Nth
Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = \paren {1 + \dfrac 1 n}^n$. Then $\sequence {x_n}$ converges to a limit as $n$ increases without bound.
Between two Real Numbers exists Rational Number
Let $a, b \in \R$ be real numbers such that $a < b$. Then: : $\exists r \in \Q: a < r < b$
Power over Factorial
Let $x \in \R: x > 0$ be a positive real number. Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = \dfrac {x^n} {n!}$. Then $\sequence {x_n}$ converges to zero.
Sequence of Powers of Reciprocals is Null Sequence
Let $r \in \Q_{>0}$ be a strictly positive rational number. Let $\sequence {x_n}$ be the sequence in $\R$ defined as: : $x_n = \dfrac 1 {n^r}$ Then $\sequence {x_n}$ is a null sequence.
Euler's Number: Limit of Sequence implies Limit of Series
Let Euler's number $e$ be defined as: :$\displaystyle e := \lim_{n \to \infty} \left({1 + \frac 1 n}\right)^n$ Then: :$\displaystyle e = \sum_{k \mathop \ge 0} \frac 1 {k!}$ That is: :$\displaystyle e = \frac 1 {0!} + \frac 1 {1!} + \frac 1 {2!} + \frac 1 {3!} + \frac 1 {4!} \cdots$
Degree of Field Extensions is Multiplicative
Let $E, K$ and $F$ be fields. Let $E / K$ and $K / F$ be finite field extensions. Then $E / F$ is a finite field extension, and: :$\index E F = \index E K \index K F$ where $\index E F$ denotes the degree of $E / F$
Convergent Sequence in Metric Space has Unique Limit
Let $\left({X, d}\right)$ be a metric space. Let $\left \langle {x_n} \right \rangle$ be a sequence in $\left({X, d}\right)$. Then $\left \langle {x_n} \right \rangle$ can have at most one limit.
Lower and Upper Bounds for Sequences
Let $\sequence {x_n}$ be a sequence in $\R$. Let $x_n \to l$ as $n \to \infty$. Then: :$(1): \quad \forall n \in \N: x_n \ge a \implies l \ge a$ :$(2): \quad \forall n \in \N: x_n \le b \implies l \le b$
Ring Homomorphism from Division Ring is Monomorphism or Zero Homomorphism
Let $\struct {R, +_R, \circ}$ and $\struct {S, +_S, *}$ be rings whose zeros are $0_R$ and $0_S$ respectively. Let $\phi: R \to S$ be a ring homomorphism. If $R$ is a division ring, then either: : $(1): \quad \phi$ is a monomorphism (that is, $\phi$ is injective) : $(2): \quad \phi$ is the zero homomorphism (that is, $\forall a \in R: \map \phi a = 0_S$).
Kernel of Ring Epimorphism is Ideal
Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring epimorphism. Then: :The kernel of $\phi$ is an ideal of $R_1$. :There is a unique ring isomorphism $g: R_1 / K \to R_2$ such that: ::$g \circ q_K = \phi$ :$\phi$ is a ring isomorphism  $K = \set {0_{R_1} }$.
Unbounded Monotone Sequence Diverges to Infinity
Let $\sequence {x_n}$ be a sequence in $\R$. Let $\sequence {x_n}$ be monotone, that is either increasing or decreasing.
Reciprocal of Null Sequence
Let $\sequence {x_n}$ be a sequence in $\R$. Let $\forall n \in \N: x_n > 0$. Then: :$x_n \to 0$ as $n \to \infty$  $\size {\dfrac 1 {x_n} } \to \infty$ as $n \to \infty$
Modulus of Limit
Let $X$ be one of the standard number fields $\Q, \R, \C$. Let $\sequence {x_n}$ be a sequence in $X$. Let $\sequence {x_n}$ be convergent to the limit $l$. That is, let $\displaystyle \lim_{n \mathop \to \infty} x_n = l$. Then :$\displaystyle \lim_{n \mathop \to \infty} \cmod {x_n} = \cmod l$ where $\cmod {x_n}$ is the modulus of $x_n$.
Strictly Increasing Sequence of Natural Numbers
Let $\N_{>0}$ be the set of natural numbers without zero: : $\N_{>0} = \left\{{1, 2, 3, \ldots}\right\}$ Let $\left \langle {n_r} \right \rangle$ be strictly increasing sequence in $\N_{>0}$. Then: : $\forall r \in \N_{>0}: n_r \ge r$
Limit of Subsequence equals Limit of Sequence
Let $T = \struct {S, \tau}$ be a topological space. Let $\sequence {x_n}$ be a sequence in $T$. Let $l \in S$ such that: :$\displaystyle \lim_{n \mathop \to \infty} x_n = l$ Let $\sequence {x_{n_r} }$ be a subsequence of $\sequence {x_n}$. Then: :$\displaystyle \lim_{r \mathop \to \infty} x_{n_r} = l$ That is, the limit of a convergent sequence in a topological space equals the limit of any subsequence of it.
Root of Number Greater than One
Let $x \in \R$ be a real number. Let $n \in \N^*$ be a natural number such that $n > 0$. Then $x \ge 1 \implies x^{1/n} \ge 1$ where $x^{1/n}$ is the $n$th root of $x$.
Limit of Root of Positive Real Number
Let $x \in \R: x > 0$ be a real number. Let $\sequence {x_n}$ be the sequence in $\R$ defined as: :$x_n = x^{1 / n}$ Then $x_n \to 1$ as $n \to \infty$.
Hero's Method
Let $a \in \R$ be a real number such that $a > 0$. Let $x_1 \in \R$ be a real number such that $x_1 > 0$. Let $\sequence {x_n}$ be the sequence in $\R$ defined recursively by: :$\forall n \in \N_{>0}: x_{n + 1} = \dfrac {x_n + \dfrac a {x_n} } 2$ Then $x_n \to \sqrt a$ as $n \to \infty$.
Limit of Integer to Reciprocal Power
Let $\sequence {x_n}$ be the real sequence defined as $x_n = n^{1/n}$, using exponentiation. Then $\sequence {x_n}$ converges with a limit of $1$.
Difference Between Adjacent Square Roots Converges
Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = \sqrt {n + 1} - \sqrt n$. Then $\sequence {x_n}$ converges to a zero limit.
Peak Point Lemma
Let $\sequence {x_n}$ be a sequence in $\R$ which is infinite. Then $\sequence {x_n}$ has an infinite subsequence which is monotone.
Bolzano-Weierstrass Theorem
Every bounded sequence of real numbers has a convergent subsequence.
Existence of Maximum and Minimum of Bounded Sequence
Let $\sequence {x_n}$ be a bounded sequence in $\R$ (which may or may not be convergent). Let $L$ be the set of all real numbers which are the limit of some subsequence of $\sequence {x_n}$. Then $L$ has both a maximum and a minimum.
Interval Defined by Absolute Value
Let $\xi, \delta \in \R$ be real numbers. Let $\delta > 0$. Then:
Difference of Two Powers
Let $\mathbb F$ denote one of the standard number systems, that is $\Z$, $\Q$, $\R$ and $\C$. Let $n \in \N$ such that $n \ge 2$. Then for all $a, b \in \mathbb F$: {{begin-eqn}} {{eqn | l = a^n - b^n       | r = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j       | c =  }} {{eqn | r = \paren {a - b} \paren {a^{n - 1} + a^{n - 2} b + a^{n - 3} b^2 + \dotsb + a b^{n - 2} + b^{n - 1} }       | c =  }} {{end-eqn}}
Polynomial Factor Theorem
Let $P \paren x$ be a polynomial in $x$ over a field $K$ of degree $n$. Then: :$\xi \in K: P \paren \xi = 0 \iff \map P x = \paren {x - \xi} \map Q x$ where $Q$ is a polynomial of degree $n - 1$. Hence, if $\xi_1, \xi_2, \ldots, \xi_n \in K$ such that all are different, and $\map P {\xi_1} = \map P {\xi_2} = \dotsb = \map P {\xi_n} = 0$, then: :$\displaystyle \map P x = k \prod_{j \mathop = 1}^n \paren {x - \xi_j}$ where $k \in K$.
Telescoping Series/Example 1
Let $\left \langle {b_n} \right \rangle$ be a sequence in $\R$. Let $\left \langle {a_n} \right \rangle$ be a sequence whose terms are defined as: :$a_k = b_k - b_{k + 1}$ Then: :$\displaystyle \sum_{k \mathop = 1}^n a_k = b_1 - b_{n + 1}$
Terms in Convergent Series Converge to Zero
Let $\sequence {a_n}$ be a sequence in any of the standard number fields $\Q$, $\R$, or $\C$. Suppose that the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges in any of the standard number fields $\Q$, $\R$, or $\C$. Then: :$\displaystyle \lim_{n \mathop \to \infty} a_n = 0$ {{expand|Expand (on a different page) to Banach spaces}}
Linear Combination of Convergent Series
Let $\sequence {a_n}_{n \mathop \ge 1}$ and $\sequence {b_n}_{n \mathop \ge 1}$ be sequences of real numbers. Let the two series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ and $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ converge to $\alpha$ and $\beta$ respectively. Let $\lambda, \mu \in \R$ be real numbers. Then the series $\displaystyle \sum_{n \mathop = 1}^\infty \paren {\lambda a_n + \mu b_n}$ converges to $\lambda \alpha + \mu \beta$.
Abel's Theorem
Let $\displaystyle \sum_{k \mathop = 0}^\infty a_k$ be a convergent series in $\R$. Then: :$\displaystyle \lim_{x \mathop \to 1^-} \paren {\sum_{k \mathop = 0}^\infty a_k x^k} = \sum_{k \mathop = 0}^\infty a_k$ where $\displaystyle \lim_{x \mathop \to 1^-}$ denotes the limit from the left.
Alternating Series Test
Let $\sequence {a_n}_{N \mathop \ge 0}$ be a decreasing sequence of positive terms in $\R$ which converges with a limit of zero. That is, let $\forall n \in \N: a_n \ge 0, a_{n + 1} \le a_n, a_n \to 0$ as $n \to \infty$ Then the series: :$\displaystyle \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} a_n = a_1 - a_2 + a_3 - a_4 + \dotsb$  converges.
Euler's Identity
:$e^{i \pi} + 1 = 0$
Intermediate Value Theorem
Let $f: S \to \R$ be a real function on some subset $S$ of $\R$. Let $I \subseteq S$ be a real interval. Let $f: I \to \R$ be continuous on $I$. Let $a, b \in I$. Let $k \in \R$ lie between $\map f a$ and $\map f b$. That is, either: :$\map f a < k < \map f b$ or: :$\map f b < k < \map f a$ Then $\exists c \in \openint a b$ such that $\map f c = k$.
Tail of Convergent Series tends to Zero
Let $\sequence {a_n}_{n \mathop \ge 1}$ be a sequence of real numbers. Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a convergent series. Let $N \in \N_{\ge 1}$ be a natural number. Let $\displaystyle \sum_{n \mathop = N}^\infty a_n$ be the tail of the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$. Then: :$\displaystyle \sum_{n \mathop = N}^\infty a_n$ is convergent :$\displaystyle \sum_{n \mathop = N}^\infty a_n \to 0$ as $N \to \infty$. That is, the tail of a convergent series tends to zero.
Comparison Test
Let $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ be a convergent series of positive real numbers. Let $\sequence {a_n}$ be a sequence $\R$ or sequence in $\C$. Let $\forall n \in \N_{>0}: \cmod {a_n} \le b_n$. Then the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges absolutely.
Ratio Test
Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series of real numbers in $\R$, or a series of complex numbers in $\C$. Let the sequence $\sequence {a_n}$ satisfy: :$\displaystyle \lim_{n \mathop \to \infty} \size {\frac {a_{n + 1} } {a_n} } = l$ :If $l > 1 $, then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ diverges. :If $l < 1 $, then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges absolutely.
Limit of Subsequence of Bounded Sequence
Let $\sequence {x_n}$ be a sequence in $\R$. Let $\sequence {x_n}$ be bounded. Let $b \in \R$ be a real number. Suppose that $\forall N: \exists n > N: x_n \ge b$. Then $\sequence {x_n}$ has a subsequence which converges to a limit $l \ge b$.
Terms of Bounded Sequence Within Bounds
Let $\sequence {x_n}$ be a sequence in $\R$. Let $\sequence {x_n}$ be bounded. Let the limit superior of $\sequence {x_n}$ be $\overline l$. Let the limit inferior of $\sequence {x_n}$ be $\underline l$. Then: :$\forall \epsilon > 0: \exists N: \forall n > N: x_n < \overline l + \epsilon$ :$\forall \epsilon > 0: \exists N: \forall n > N: x_n > \underline l - \epsilon$
Convergence of Limsup and Liminf
Let $\sequence {x_n}$ be a sequence in $\R$. Let the limit superior of $\sequence {x_n}$ be $\overline l$. Let the limit inferior of $\sequence {x_n}$ be $\underline l$. Then $\left \langle {x_n} \right \rangle$ converges to a limit $l$  $\overline l = \underline l = l$. Hence a bounded real sequence converges  all its convergent subsequences have the same limit.
Limsup and Liminf are Limits of Bounds
Let $\sequence {x_n}$ be a sequence in $\R$. Let $\sequence {x_n}$ be bounded. Let $\displaystyle \overline l = \limsup_{n \mathop \to \infty} x_n$ be the limit superior and $\displaystyle \liminf_{n \mathop \to \infty} x_n$ the limit inferior of $\sequence {x_n}$. Then: :$\displaystyle \overline l = \limsup_{n \mathop \to \infty} x_n = \map {\lim_{n \mathop \to \infty} } {\sup_{k \mathop \ge n} x_k}$ :$\displaystyle \underline l = \liminf_{n \mathop \to \infty} x_n = \map {\lim_{n \mathop \to \infty} } {\inf_{k \mathop \ge n} x_k}$
Nth Root Test
Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series of real numbers $\R$ or complex numbers $\C$. Let the sequence $\sequence {a_n}$ be such that the limit superior $\displaystyle \limsup_{n \mathop \to \infty} \size {a_n}^{1/n} = l$. Then: :If $l > 1$, the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ diverges. :If $l < 1$, the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges absolutely.
Series of Power over Factorial Converges
The series $\displaystyle \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$ converges for all real values of $x$.
Area of Triangle in Terms of Side and Altitude
The area of a triangle $\triangle ABC$ is given by: :$\dfrac {c \cdot h_c} 2 = \dfrac {b \cdot h_b} 2 = \dfrac {a \cdot h_a} 2$ where: :$a, b, c$ are the sides :$h_a, h_b, h_c$ are the altitudes from $A$, $B$ and $C$ respectively.
Absolutely Convergent Series is Convergent
Let $V$ be a Banach space with norm $\norm {\, \cdot \,}$. Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be an absolutely convergent series in $V$. Then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is convergent.
Limit Comparison Test
Let $\left \langle {a_n} \right \rangle$ and $\left \langle {b_n} \right \rangle$ be sequences in $\R$. Let $\displaystyle \frac {a_n}{b_n} \to l$ as $n \to \infty$ where $l \in \R_{>0}$. Then the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ and $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ are either both convergent or both divergent.
Area of Parallelogram
The area of a parallelogram equals the product of one of its bases and the associated altitude.
Area of Triangle in Terms of Inradius and Exradii
The area of a $\triangle ABC$ is given by the formula: :$(ABC) = \rho_a \left({s - a}\right) = \rho_b \left({s - b}\right) = \rho_c \left({s - c}\right) = \rho s = \sqrt {\rho_a \rho_b \rho_c \rho}$ where: :$s$ is the semiperimeter :$I$ is the incenter :$\rho$ is the inradius :$I_a, I_b, I_c$ are the excenters :$\rho_a, \rho_b, \rho_c$ are the exradii from $I_a, I_b, I_c$, respectively.
Area of Triangle in Terms of Circumradius
Let $\triangle ABC$ be a triangle whose sides are of lengths $a, b, c$. Then the area $\AA$ of $\triangle ABC$ is given by: :$\AA = \dfrac {a b c} {4 R}$ where $R$ is the circumradius of $\triangle ABC$.
Stewart's Theorem
Let $\triangle ABC$ be a triangle with sides $a, b, c$. Let $CP$ be a cevian from $C$ to $P$. :400px Then: :$a^2 \cdot AP + b^2 \cdot PB = c \paren {CP^2 + AP \cdot PB}$
Length of Median of Triangle
Let $\triangle ABC$ be a triangle. Let $CD$ be the median of $\triangle ABC$ which bisects $AB$. :400px The length $m_c$ of $CD$ is given by: :${m_c}^2 = \dfrac {a^2 + b^2} 2 - \dfrac {c^2} 4$
Length of Angle Bisector
Let $\triangle ABC$ be a triangle. Let $AD$ be the angle bisector of $\angle BAC$ in $\triangle ABC$. :300px Let $d$ be the length of $AD$. Then $d$ is given by: :$d^2 = \dfrac {b c} {\paren {b + c}^2} \paren {\paren {b + c}^2 - a^2}$ where $a$, $b$, and $c$ are the sides opposite $A$, $B$ and $C$ respectively.
Supremum of Subset
Let $\left({U, \preceq}\right)$ be an ordered set. Let $S \subseteq U$. Let $T \subseteq S$. Let $S$ admit a supremum (in $U$). If $T$ also admits a supremum (in $U$), then $\sup \left({T}\right) \preceq\sup \left({S}\right)$.
Area of Triangle
This page gathers a variety of formulas for the area of a triangle.
Intersecting Chord Theorem
Let $AC$ and $BD$ both be chords of the same circle. Let $AC$ and $BD$ intersect at $E$. Then $AE \cdot EC = DE \cdot EB$.  :''If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.'' {{EuclidPropRef|III|35}}
Supremum Plus Constant
Let $S$ be a subset of the set of real numbers $\R$. Let $S$ be bounded above. Let $\xi \in \R$. Then: :$\displaystyle \map {\sup_{x \mathop \in S} } {x + \xi} = \xi + \map {\sup_{x \mathop \in S} } x$ where $\sup$ denotes supremum.
Suprema and Infima of Combined Bounded Functions
Let $f$ and $g$ be real functions. Let $c$ be a constant.
Multiple of Infimum
Let $T \subseteq \R: T \ne \varnothing$ be a non-empty subset of the set of real numbers $\R$. Let $T$ be bounded below. Let $z \in \R: z > 0$ be a (strictly) positive real number. Then: :$\displaystyle \map {\inf_{x \mathop \in T} } {z x} = z \, \map {\inf_{x \mathop \in T} } x$ where $\inf$ denotes infimum.
Law of Sines
For any triangle $\triangle ABC$: :$\dfrac a {\sin A} = \dfrac b {\sin B} = \dfrac c {\sin C} = 2 R$ where: :$a$, $b$, and $c$ are the sides opposite $A$, $B$ and $C$ respectively :$R$ is the circumradius of $\triangle ABC$.
Negative of Infimum is Supremum of Negatives
Let $T$ be a non-empty subset of the real numbers $\R$. Let $T$ be bounded below. Then: : $(1): \quad \set {x \in \R: -x \in T}$ is bounded above : $(2): \quad \displaystyle -\inf_{x \mathop \in T} x = \map {\sup_{x \mathop \in T} } {-x}$ where $\sup$ and $\inf$ denote the supremum and infimum respectively.
Infimum Plus Constant
Let $T$ be a subset of the set of real numbers. Let $T$ be bounded below. Let $\xi \in \R$. Then: :$\displaystyle \map {\inf_{x \mathop \in T} } {x + \xi} = \xi + \map {\inf_{x \mathop \in T} } x$ where $\inf$ denotes infimum.
Infimum of Subset
Let $\struct {U, \preceq}$ be an ordered set. Let $S \subseteq U$. Let $T \subseteq S$. Let $\struct {S, \preceq}$ admit an infimum in $U$. If $T$ also admits an infimum in $U$, then $\map \inf S \preceq \map \inf T$.
Construction of Equilateral Triangle
On a given straight line segment, it is possible to construct an equilateral triangle. {{:Euclid:Proposition/I/1}}
Construction of Equal Straight Line
At a given point, it is possible to construct a straight line segment of length equal to that of any given straight line segment. The given point will be an endpoint of the constructed straight line segment. {{:Euclid:Proposition/I/2}}
Construction of Equal Straight Lines from Unequal
Given two unequal straight line segments, it is possible to cut off from the greater a straight line segment equal to the lesser. {{:Euclid:Proposition/I/3}}
Triangle Side-Angle-Side Equality
If $2$ triangles have: : $2$ sides equal to $2$ sides respectively : the angles contained by the equal straight lines equal they will also have: : their third sides equal : the remaining two angles equal to their respective remaining angles, namely, those which the equal sides subtend.
Isosceles Triangle has Two Equal Angles
In isosceles triangles, the angles at the base are equal to each other. Also, if the equal straight lines are extended, the angles under the base will also be equal to each other. {{:Euclid:Proposition/I/5}}
Triangle with Two Equal Angles is Isosceles
If a triangle has two angles equal to each other, the sides which subtend the equal angles will also be equal to one another. Hence, by definition, such a triangle will be isosceles. {{:Euclid:Proposition/I/6}}
Two Lines Meet at Unique Point
Let two straight line segments be constructed on a straight line segment from its endpoints so that they meet at a point. Then there cannot be two other straight line segments equal to the former two respectively, constructed on the same straight line segment and on the same side of it, meeting at a different point. {{:Euclid:Proposition/I/7}}
Triangle Side-Side-Side Equality
Let two triangles have all $3$ sides equal. Then they also have all $3$ angles equal. Thus two triangles whose sides are all equal are themselves congruent.
Bisection of Angle
It is possible to bisect any given rectilineal angle. {{:Euclid:Proposition/I/9}}
Bisection of Straight Line
It is possible to bisect a straight line segment. {{:Euclid:Proposition/I/10}}