hcju/proofwikips huggingface dataset:
External Angle of Triangle Greater than Internal Opposite
The external angle of a triangle is greater than either of the opposite internal angles. {{:Euclid:Proposition/I/16}}
Limit iff Limits from Left and Right
Let $f$ be a real function defined on an open interval $\openint a b$ except possibly at a point $c \in \openint a b$. Then: :$\map f x \to l$ as $x \to c$ : :$\map f x \to l$ as $x \to c^-$ and :$\map f x \to l$ as $x \to c^+$
Limit of Function by Convergent Sequences
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $S \subseteq A_1$ be an open set of $M_1$. Let $f$ be a mapping defined on $S$, except possibly at the point $c \in S$. Then $\displaystyle \lim_{x \mathop \to c} f \left({x}\right) = l$ iff: : for each sequence $\left \langle {x_n} \right \rangle$ of points of $S$ such that $\forall n \in \N_{>0}: x_n \ne c$ and $\displaystyle \lim_{n \to \infty} x_n = c$ it is true that: : $\displaystyle \lim_{n \to \infty} f \left({x_n}\right) = l$
Combination Theorem for Limits of Functions
Let $X$ be one of the standard number fields $\Q, \R, \C$. Let $f$ and $g$  be functions defined on an open subset $S \subseteq X$, except possibly at the point $c \in S$. Let $f$ and $g$ tend to the following limits: :$\displaystyle \lim_{x \to c} \ f \left({x}\right) = l$ :$\displaystyle \lim_{x \to c} \ g \left({x}\right) = m$ Let $\lambda, \mu \in X$ be arbitrary numbers in $X$. Then the following results hold:
Real Polynomial Function is Continuous
A (real) polynomial function is continuous at every point. Thus a (real) polynomial function is continuous on every interval of $\R$.
Linear Function is Continuous
Let $\alpha, \beta \in \R$ be real numbers. Let $f : \R \to \R$ be the real function with: :$\map f x = \alpha x + \beta$ for all $x \in \R$. Then $f$ is continuous at every real number $c \in \R$.
Real Rational Function is Continuous
A real rational function is continuous at every point at which it is defined. Thus a real rational function is continuous on every interval of $\R$ not containing a root of the denominator of the function.
Limit of Composite Function
Let $f$ and $g$ be real functions. Let: :$\displaystyle \lim_{y \mathop \to \eta} \map f y = l$ :$\displaystyle \lim_{x \mathop \to \xi} \map g x = \eta$ Then, if either: :'''Hypothesis 1:''' $f$ is continuous at $\eta$ (that is $l = \map f \eta$) or: :'''Hypothesis 2:''' for some open interval $I$ containing $\xi$, it is true that $\map g x \ne \eta$ for any $x \in I$ except possibly $x = \xi$ then: : $\displaystyle \lim_{x \mathop \to \xi} \map f {\map g x} = l$
Limit of Image of Sequence
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Let $f: A_1 \to A_2$ be a mapping which is continuous at $a \in A_1$. Let $\sequence {x_n}$ be a sequence of points in $A_1$ such that: :$\displaystyle \lim_{n \mathop \to \infty} x_n = a$ where $\displaystyle \lim_{n \mathop \to \infty} x_n$ is the limit of $x_n$. Then: :$\displaystyle \lim_{n \mathop \to \infty} \map f {x_n} = \map f a$ That is: :$\displaystyle \lim_{n \mathop \to \infty} \map f {x_n} = \map f {\lim_{n \mathop \to \infty} x_n}$ That is, for a continuous mapping, the limit and function symbols commute.
Image of Interval by Continuous Function is Interval
Let $I$ be a real interval. Let $f: I \to \R$ be a continuous real function. Then the image of $f$ is a real interval.
Interval Divided into Subsets
Let $\mathbb I$ be a real interval. Let $S$ and $T$ be non-empty subsets of $\mathbb I$ such that $\mathbb I \subseteq S \cup T$. Then one of $S$ or $T$ contains an element at zero distance from the other.
Distance from Subset of Real Numbers
Let $S$ be a subset of the set of real numbers $\R$. Let $x \in \R$ be a real number. Let $\map d {x, S}$ be the distance between $x$ and $S$. Then:
Limit of Sequence to Zero Distance Point
Let $S$ be a non-empty subset of $\R$. Let the distance $\map d {\xi, S} = 0$ for some $\xi \in \R$. Then there exists a sequence $\sequence {x_n}$ in $S$ such that $\displaystyle \lim_{n \mathop \to \infty} x_n = \xi$.
Image of Closed Real Interval is Bounded
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$. Then $f$ is bounded on $\closedint a b$.
Convergent Subsequence in Closed Interval
Let $\closedint a b$ be a closed real interval. Then every sequence of points of $\closedint a b$ contains a subsequence which converges to a point in $\closedint a b$.
Max and Min of Function on Closed Real Interval
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$. Then $f$ reaches a maximum and a minimum on $\closedint a b$.
Continuous Image of Closed Interval is Closed Interval
Let $f$ be a real function which is continuous on the closed interval $\left[{a \,.\,.\, b}\right]$. Then the image of $\left[{a \,.\,.\, b}\right]$ under $f$ is also a closed interval.
Retraction Theorem
Let $M$ be a compact manifold with boundary $\partial M$. Then there is no smooth mapping $f: M \to \partial M$ such that $\partial f: \partial M \to \partial M$ is the identity.
Classification of Compact One-Manifolds
Every compact one-dimensional manifold is diffeomorphic to either a circle or a closed interval.
Differentiable Function is Continuous
Let $f$ be a real function defined on an interval $I$. Let $x_0 \in I$ such that $f$ is differentiable at $x_0$. Then $f$ is continuous at $x_0$.
Quotient Rule for Derivatives
Let $\map j x, \map k x$ be real functions defined on the open interval $I$. Let $\xi \in I$ be a point in $I$ at which both $j$ and $k$ are differentiable. Define the real function $f$ on $I$ by: :$\displaystyle \map f x = \begin{cases} \dfrac {\map j x} {\map k x} & : \map k x \ne 0 \\ 0 & : \text{otherwise} \end{cases}$ Then, if $\map k \xi \ne 0$, $f$ is differentiable at $\xi$, and furthermore: :$\map {f'} \xi = \dfrac {\map {j'} \xi \map k \xi - \map j \xi \map {k'} \xi} {\paren {\map k \xi}^2}$ It follows from the definition of derivative that if $j$ and $k$ are both differentiable on the interval $I$, then: :$\displaystyle \forall x \in I: \map k x \ne 0 \implies \map {f'} x = \frac {\map {j'} x \map k x - \map j x \map {k'} x} {\paren {\map k x}^2}$
Preimage Theorem
Let $y$ be a regular value of a smooth submersion $f:X \to Y$. Then the preimage $f^{-1}(y)$ is a smooth submanifold of $X$, with $\dim f^{-1}(y) = \dim X - \dim Y$.
Backwards Induction
Let $P$ be a propositional function on the natural numbers $\N$. Suppose that: :$(1): \quad \forall n \in \N: \map P {2^n}$ holds. :$(2): \quad \map P n \implies \map P {n - 1}$. Then $\map P n$ holds for all $\forall n \in \N$. The proof technique based on this result is called '''backwards induction'''.
Inequalities Concerning Roots
Let $\closedint X Y$ be a closed real interval such that $0 < X \le Y$. Let $x, y \in \closedint X Y$. Then: :$\forall n \in \N_{> 0}: X Y^{1/n} \size {x - y} \le n X Y \size {x^{1/n} - y^{1/n} } \le Y X^{1/n} \size {x - y}$
Continuity of Root Function
Let $n \in \N_{>0}$ be a non-zero natural number. Let $f: \hointr 0 \infty \to \R$ be the real function defined by $\map f x = x^{1/n}$. Then $f$ is continuous at each $\xi > 0$ and continuous on the right at $\xi = 0$.
P-adic Norm is Norm
The $p$-adic norm forms a norm on the rational numbers $\Q$.
Fundamental Theorem of Algebra
Every non-constant polynomial with coefficients in $\C$ has a root in $\C$.
Extendability Theorem for Intersection Numbers
Let $X = \partial W$ be a smooth manifold which is the boundary of a smooth compact manifold $W$. Let $Y$ be a smooth manifold, $Z$ be a closed smooth submanifold of $Y$, and $f: X \to Y$ a smooth map. Let there exist a smooth map $g: W \to Y$ such that $g \restriction_X = f$. Then: :$I \left({f, Z}\right) = 0$ where $I \left({f, Z}\right)$ is the intersection number. {{explain|what $I \left({f, Z}\right)$ is the intersection number of: presumably the words will go something like "... the intersection number of $f$ with respect to $Z$", or something.}}
Homotopy Group is Group
The set of all homotopy classes of continuous mappings: :$c: \closedint 0 1^n \to X$ satisfying: :$\map c {\partial \closedint 0 1^n} = x_0$ in a space $X$ at a base point $x_0$, under the operation of concatenation on class members, forms a group. {{finish|This operation needs to be shown well-defined; probably a nasty proof}} This group is called the $n$th homotopy group.
Derivative at Maximum or Minimum
Let $f$ be a real function which is differentiable on the open interval $\openint a b$. Let $f$ have a local minimum or local maximum at $\xi \in \openint a b$. Then: :$\map {f'} \xi = 0$
Behaviour of Function Near Limit
Let $f$ be a real function. Let $f \left({x}\right) \to l$ as $x \to \xi$. Then: * If $l > 0$, then $\exists h > 0: \forall x: \xi - h < x < \xi + h, x \ne \xi: f \left({x}\right) > 0$ * If $l < 0$, then $\exists h > 0: \forall x: \xi - h < x < \xi + h, x \ne \xi: f \left({x}\right) < 0$
Fundamental Group is Independent of Base Point for Path-Connected Space
Let $X$ be a path-connected space. For $x \in X$ let $\pi_1(X,x)$ denote the fundamental group. For $x, y \in X$, there is an isomorphism: :$\phi: \pi_1 \left({X, x}\right) \to \pi_1 \left({X, y}\right)$
List of Fundamental Groups for 2-Manifolds
For the following two-manifolds, the fundamental group for any point in $X$, written $\pi_1 \left({X}\right)$ is isomorphic to the listed group: :$\pi_1 \left({\Bbb S^1 \times \left[{0\,.\,.\,1}\right]}\right) = \Z$ :$\pi_1 \left({\Bbb S^2}\right) = \left\{{e}\right\}$, the trivial group. :$\pi_1 \left({\Bbb T^2}\right) = \pi_1 \left({\Bbb S^1 \times \Bbb S^1}\right) = \Z \times \Z$ :$\pi_1 \left({\Bbb {RP}^2}\right) = \Z_2$
Rolle's Theorem
Let $f$ be a real function which is: :continuous on the closed interval $\closedint a b$ and: :differentiable on the open interval $\openint a b$. Let $\map f a = \map f b$. Then: :$\exists \xi \in \openint a b: \map {f'} \xi = 0$
Mean Value Theorem
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$. Then: :$\exists \xi \in \openint a b: \map {f'} \xi = \dfrac {\map f b - \map f a} {b - a}$
Zero Derivative implies Constant Function
Let $f$ be a real function which is continuous on the closed interval $\left[{a \,.\,.\, b}\right]$ and differentiable on the open interval $\left({a \,.\,.\, b}\right)$. Suppose that: :$\forall x \in \left({a \,.\,.\, b}\right): f' \left({x}\right) = 0$ Then $f$ is constant on $\left[{a \,.\,.\, b}\right]$.
Ostrowski's Theorem
Every non-trivial norm on the rational numbers $\Q$ is equivalent to either: :the $p$-adic norm $\norm {\, \cdot \,}_p$ for some prime $p$ or: :the absolute value, $\size {\, \cdot \,}$.
Cauchy Mean Value Theorem
Let $f$ and $g$ be a real functions which are continuous on the closed interval $\left[{a \,.\,.\, b}\right]$ and differentiable on the open interval $\left({a \,.\,.\, b}\right)$. Suppose that: :$\forall x \in \left({a \,.\,.\, b}\right): g' \left({x}\right) \ne 0$ Then: :$\exists \xi \in \left({a \,.\,.\, b}\right): \dfrac {f' \left({\xi}\right)} {g' \left({\xi}\right)} = \dfrac {f \left({b}\right) - f \left({a}\right)} {g \left({b}\right) - g \left({a}\right)}$
Homology Group is Group
The $p^{th}$ singular homology group of a space $X$ is a group.
Limit of Absolute Value
Let $x, \xi \in \R$ be real numbers. Then: :$\left\vert{x - \xi}\right\vert \to 0$ as $x \to \xi$ where $\left\vert{x - \xi}\right\vert$ denotes the Absolute Value.
Limit of Function in Interval
Let $f$ be a real function which is defined on the open interval $\openint a b$. Let $\xi \in \openint a b$ Suppose that, $\forall x \in \openint a b$, either: :$\xi \le \map f x \le x$ or: :$x \le \map f x \le \xi$ Then $\map f x \to \xi$ as $x \to \xi$.
Derivative of Monotone Function
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.
Strictly Monotone Real Function is Bijective
Let $f$ be a real function which is defined on $I \subseteq \R$. Let $f$ be strictly monotone on $I$. Let the image of $f$ be $J$.  Then $f: I \to J$ is a bijection.
Inverse of Strictly Monotone Function
Let $f$ be a real function which is defined on $I \subseteq \R$. Let $f$ be strictly monotone on $I$. Let the image of $f$ be $J$.  Then $f$ always has an inverse function $f^{-1}$ and: : if $f$ is strictly increasing then so is $f^{-1}$ : if $f$ is strictly decreasing then so is $f^{-1}$.
Rokhlin's Theorem (Intersection Forms)
Let $M$ be a smooth 4-manifold. Then: :$\map {\omega_2} {\map T M} = 0 \implies \operatorname {sign} Q_M = 0 \pmod {16}$ where: :$Q_M$ is the intersection form :$\map T M$ is the tangent bundle :$\omega_2$ is the second Stiefel-Whitney class.
Convex Real Function is Continuous
Let $f$ be a real function which is convex on the open interval $\left({a \,.\,.\, b}\right)$. Then $f$ is continuous on $\left({a \,.\,.\, b}\right)$.
Convex Real Function is Left-Hand and Right-Hand Differentiable
Let $f$ be a real function which is either convex on the open interval $\left({a \,.\,.\, b}\right)$. Then the left-hand derivative $f'_- \left({x}\right)$ and right-hand derivative $f'_+ \left({x}\right)$ both exist for all $x \in \left({a \,.\,.\, b}\right)$.
Real Function is Convex iff Derivative is Increasing
Let $f$ be a real function which is differentiable on the open interval $\openint a b$. Then: :$f$ is convex on $\openint a b$ : :its derivative $f'$ is increasing on $\openint a b$. Thus the intuitive result that a convex function "gets steeper".
Inverse of Strictly Increasing Convex Real Function is Concave
Let $f$ be a real function which is convex on the open interval $I$. Let $J = f \left[{I}\right]$. If $f$ be strictly increasing on $I$, then $f^{-1}$ is concave on $J$.
Upper and Lower Bounds of Integral
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$. Let $\displaystyle \int_a^b \map f x \rd x$ be the definite integral of $\map f x$ over $\closedint a b$. Then: :$\displaystyle m \paren {b - a} \le \int_a^b \map f x \rd x \le M \paren {b - a}$ where: :$M$ is the maximum of $f$ :$m$ is the minimum of $f$ on $\closedint a b$.
Wilson's Theorem
A (strictly) positive integer $p$ is a prime : :$\paren {p - 1}! \equiv -1 \pmod p$
Integral of Constant
Let $c$ be a constant.
Sum of Integrals on Adjacent Intervals for Continuous Functions
Let $f$ be a real function which is continuous on any closed interval $I$. Let $a, b, c \in I$. Then: :$\ds \int_a^c \map f t \rd t + \int_c^b \map f t \rd t = \int_a^b \map f t \rd t$
Primitives which Differ by Constant
Let $F$ be a primitive for a real function $f$ on the closed interval $\closedint a b$. Let $G$ be a real function defined on $\closedint a b$. Then $G$ is a primitive for $f$ on $\closedint a b$ : :$\exists c \in \R: \forall x \in \closedint a b: \map G x = \map F x + c$ That is,  $F$ and $G$ differ by a constant on the whole interval.
Definite Integral of Function plus Constant
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$. Let $c$ be a constant. Then: :$\ds \int_a^b \paren {\map f t + c} \rd t = \int_a^b \map f t \rd t + c \paren {b - a}$
Continuous Real Function is Darboux Integrable
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$. Then $f$ is Darboux integrable on $\closedint a b$.
Definite Integral on Zero Interval
Let $f$ be a real function which is defined on the closed interval $\Bbb I := \closedint a b$, where $a < b$. Then: :$\displaystyle \forall c \in \Bbb I: \int_c^c \map f t \rd t = 0$
Linear Combination of Derivatives
Let $f \left({x}\right), g \left({x}\right)$ be real functions defined on the open interval $I$. Let $\xi \in I$ be a point in $I$ at which both $f$ and $g$ are differentiable. Then: :$D \left({\lambda f + \mu g}\right) = \lambda D f + \mu D g$ at the point $\xi$. It follows from the definition of derivative that if $f$ and $g$ are both differentiable on the interval $I$, then: :$\forall x \in I: D \left({\lambda f \left({x}\right) + \mu g \left({x}\right)}\right) = \lambda D f \left({x}\right) + \mu D g \left({x}\right)$
Linear Combination of Integrals
Let $f$ and $g$ be real functions which are integrable on the closed interval $\closedint a b$. Let $\lambda$ and $\mu$ be real numbers. Then the following results hold:
Integration by Parts
Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$. Let $f$ and $g$ have primitives $F$ and $G$ respectively on $\closedint a b$. Then:
Integration by Substitution
Let $\phi$ be a real function which has a derivative on the closed interval $\closedint a b$. Let $I$ be an open interval which contains the image of $\closedint a b$ under $\phi$. Let $f$ be a real function which is continuous on $I$.
Relative Sizes of Definite Integrals
Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$, where $a < b$. If: :$\forall t \in \closedint a b: \map f t \le \map g t$ then: :$\displaystyle \int_a^b \map f t \rd t \le \int_a^b \map g t \rd t$ Similarly, if: :$\forall t \in \closedint a b: \map f t < \map g t$ then: :$\displaystyle \int_a^b \map f t \rd t < \int_a^b \map g t \rd t$
Absolute Value of Definite Integral
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$. Then: :$\displaystyle \size {\int_a^b \map f t \rd t} \le \int_a^b \size {\map f t} \rd t$
Integral Test
Let $f$ be a real function which is continuous, positive and decreasing on the interval $\hointr 1 {+\infty}$. Let the sequence $\sequence {\Delta_n}$ be defined as: :$\displaystyle \Delta_n = \sum_{k \mathop = 1}^n \map f k - \int_1^n \map f x \rd x$ Then $\sequence {\Delta_n} $ is decreasing and bounded below by zero. Hence it converges.
Properties of Natural Logarithm
Let $x \in \R$ be a real number such that $x > 0$. Let $\ln x$ be the natural logarithm of $x$. Then:
Harmonic Series is Divergent
The harmonic series: :$\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 n$  diverges.
Sum of Reciprocals of Primes is Divergent
Let $n \in \N: n \ge 1$. There exists a (strictly) positive real number $C \in \R_{>0}$ such that: :$(1): \quad \displaystyle \sum_{\substack {p \mathop \in \Bbb P \\ p \mathop \le n} } \frac 1 p > \map \ln {\ln n} - C$ where $\Bbb P$ is the set of all prime numbers. :$(2): \quad \displaystyle \lim_{n \mathop \to \infty} \paren {\map \ln {\ln n} - C} = +\infty$
Measure of Interval is Length
Let $I$ be a real interval whose endpoints are $a$ and $b$. Then $I$ is Lebesgue measurable, and the value of the measure is the length of the interval $b - a$.
Measurable Sets form Algebra of Sets
Let $\mu^*$ be an outer measure on a set $X$. Then the set of $\mu^*$-measurable sets is an algebra of sets.
Classification of Compact Three-Manifolds Supporting Zero-Curvature Geometry
Every closed, orientable, path connected $3$-dimensional Riemannian manifold which supports a geometry of zero curvature is homeomorphic to one of the following: * Torus $\mathbb T^3$ * Half-Twist Cube * Quarter-Twist Cube * Hantschze-Wendt Manifold * $\frac 1 6$-Twist Hexagonal Prism * $\frac 1 3$-Twist Hexagonal Prism The $3$-torus is described on the torus page. The other manifolds can be described using quotient spaces on familiar prisms, with the equivalence relations described below. <gallery perRow="5"> File:Halftwistcube.JPG|The Half-Twist Cube File:Hantschzewendt.JPG|The Hantschze-Wendt Manifold File:Quartertwistcube.JPG|The Quarter-Twist Cube File:Sixthtwisthexagon.JPG|The $\frac 1 6$-Twist Hexagonal Prism File:Thirdtwisthexagon.JPG|The $\frac 1 3$-Twist Hexagonal Prism </gallery>
Sum of Logarithms
{{:Sum of Logarithms/General Logarithm}}
Heine-Borel Theorem/Real Line
Let $\R$ be the real number line considered as a Euclidean space. Let $C \subseteq \R$. Then $C$ is closed and bounded in $\R$  $C$ is compact.
Measurable Image
Let $\mathfrak M$ be the set of measurable sets of $\R$. For any extended real-valued function $f: \R \to \R \cup \left\{{-\infty \,.\,.\, +\infty}\right\}$ whose domain is measurable, the following statements are equivalent: : $(1): \quad \forall \alpha \in \R: \left\{{x: f \left({x}\right) > \alpha}\right\} \in \mathfrak M$ : $(2): \quad \forall \alpha \in \R: \left\{{x: f \left({x}\right) \ge \alpha}\right\} \in \mathfrak M$ : $(3): \quad \forall \alpha \in \R: \left\{{x: f \left({x}\right) < \alpha}\right\} \in \mathfrak M$ : $(4): \quad \forall \alpha \in \R: \left\{{x: f \left({x}\right) \le \alpha}\right\} \in \mathfrak M$ These statements imply: : $(5): \quad \forall \alpha \in \R \cup \left\{{-\infty \,.\,.\, +\infty}\right\}: \left\{{x:f \left({x}\right) = \alpha}\right\} \in \mathfrak M$
Lebesgue Integral is Extension of Darboux Integral
Let $f: \closedint a b \to \R$ be a Darboux integrable function. Then it is also Lebesgue integrable, and furthermore: :$\displaystyle R \int_a^b \map f x \rd x = \int_{\closedint a b} f \rd \lambda$ where $\displaystyle R \int_a^b$ is the Darboux integral and $\displaystyle \int_{\closedint a b}$ is the Lebesgue integral.
Properties of Exponential Function
Let $x \in \R$ be a real number. Let $\exp x$ be the exponential of $x$. Then:
Euclidean Metric on Real Vector Space is Metric
The Euclidean metric on a real vector space $\R^n$ is a metric.
Metric Induces Topology
Let $M = \struct {A, d}$ be a metric space. Then the topology $\tau$ induced by the metric $d$ is a topology on $M$.
Properties of Algebras of Sets
Let $X$ be a set. Let $\mathfrak A$ be an algebra of sets on $X$. Then the following hold: :$(1): \quad$ The intersection of two sets in $\mathfrak A$ is in $\mathfrak A$. :$(2): \quad$ The difference of two sets in $\mathfrak A$ is in $\mathfrak A$. :$(3): \quad$ $X \in \mathfrak A$. :$(4): \quad$ The empty set $\O$ is in $\mathfrak A$.
Countable Sets Have Measure Zero
Let $S$ be a countable set. {{explain|Is it assumed that $S \subseteq \R$?}} Then the measure of $S$ is $\map m S = 0$.
Mean Value of Convex Real Function
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$. Let $f$ be convex on $\openint a b$. Then: :$\forall \xi \in \openint a b: \map f x - \map f \xi \ge \map {f'} \xi \paren {x - \xi}$
Upper Bound of Natural Logarithm
Let $\ln y$ be the natural logarithm of $y$ where $y \in \R_{>0}$. Then: :$\ln y \le y - 1$
Exponential of Sum/Real Numbers
Let $x, y \in \R$ be real numbers. Let $\exp x$ be the exponential of $x$. Then: :$\map \exp {x + y} = \paren {\exp x} \paren {\exp y}$
Exponential of Product
Let $x, y \in \R$ be real numbers. Let $\exp x$ be the exponential of $x$. Then: :$\map \exp {x y} = \paren {\exp y}^x$
Exponent Combination Laws
Let $a, b \in \R_{>0}$ be strictly positive real numbers. Let $x, y \in \R$ be real numbers. Let $a^x$ be defined as $a$ to the power of $x$. Then:
Derivative of Exponential at Zero
Let $\exp x$ be the exponential of $x$ for real $x$. Then: : $\displaystyle \lim_{x \mathop \to 0} \frac {\exp x - 1} x = 1$
Classification of Groups of Order up to 15
Up to isomorphism, every group of order $\order G \le 15$ is one of the below: {| class="sortable wikitable" |- bgcolor="#ececec" ! Order !! Abelian !! Non-Abelian |- |1 || $\Z_1$ ||  |- |2 || $\Z_2$ || |- |3 || $\Z_3$ || |- |4 || $\Z_4, \Z_2 \oplus \Z_2$ ||  |- |5 || $\Z_5$ || |- |6 || $\Z_6$ || $D_3 = S_3$ |-  |7 || $\Z_7$ || |- |8 || $\Z_8, \Z_4 \oplus \Z_2, \Z_2 \oplus \Z_2 \oplus \Z_2$ || $D_4, \Dic 2$ |- |9 || $\Z_9, \Z_3 \oplus \Z_3$ ||  |- |10 || $\Z_{10}$ || $D_5$ |- |11 || $\Z_{11}$ || |- |12 || $\Z_{12}, \Z_6 \oplus \Z_2$ || $D_6, A_4, \Dic 3$ |- |13 || $\Z_{13}$ || |- |14 || $\Z_{14}$ || $D_7$ |- |15 || $\Z_{15}$ ||  |} where: : $D_n$ is the dihedral group of order $2 n$ : $S_n$ is the $n$th symmetric group : $A_n$ is the alternating group on $n$ points : $\Dic 2$ is the dicyclic group of order $4 n$.
Cyclic Groups of Order p q
Let $p, q$ be primes such that $p < q$ and $p$ does not divide $q - 1$. Let $G$ be a group of order $p q$. Then $G$ is cyclic.
Derivative of Logarithm at One
Let $\ln x$ be the natural logarithm of $x$ for real $x$ where $x > 0$. Then: :$\displaystyle \lim_{x \mathop \to 0} \frac {\map \ln {1 + x} } x = 1$