hcju/proofwikips huggingface dataset:
Meet of Directed Subsets is Directed
Let $\left({S, \preceq}\right)$ be a meet semilattice. Let $D_1, D_2$ be directed subset of $S$. Then :$\left\{ {x \wedge y: x \in D_1, y \in D_2}\right\}$ is directed subset of $S$
Image of Directed Subset under Increasing Mapping is Directed
Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets. Let $f: S \to T$ be an increasing mapping. Let $D$ be a directed subset of $S$. Then :$f^\to \left({D}\right)$ is also a directed subset of $T$ where :$f^\to \left({D}\right)$ denotes the image of $D$ under $f$.
Mass of Sun from Gravitational Constant
Let the gravitational constant be known. Let the mean distance from the Earth to the sun be known. Then it is possible to calculate the mass of the sun.
Meet is Directed Suprema Preserving implies Meet of Suprema equals Supremum of Meet of Directed Subsets
Let $\struct {S, \preceq}$ be an up-complete meet semilattice. Let $\struct {S \times S, \precsim}$ be the Cartesian product of $\struct {S, \preceq}$ and $\struct {S, \preceq}$. Let $f: S \times S \to S$ be a mapping such that :$\forall s, t \in S: \map f {s, t} = s \wedge t$ and :$f$ preserves directed suprema. Let $D_1, D_2$ be directed subsets of $S$. Then :$\paren {\sup D_1} \wedge \paren {\sup D_2} = \sup \set {x \wedge y: x \in D_1, y \in D_2}$
Meet of Suprema equals Supremum of Meet of Ideals implies Ideal Supremum is Meet Preserving
Let $\mathscr S = \struct {S, \wedge, \preceq}$ be an up-complete meet semilattice. Let $f: \map {\it Ids} {\mathscr S} \to S$ be a mapping such that: :$\forall I \in \map {\it Ids} {\mathscr S}: \map f I = \sup_{\mathscr S} I$ where  :$\map {\it Ids} {\mathscr S}$ denotes the set of all ideals in $\mathscr S$ Let :$\forall I_1, I_2 \in \map {\it Ids} {\mathscr S}: \paren {\sup I_1} \wedge \paren {\sup I_2} = \sup \set {i \wedge j: i \in I_1, j \in I_2}$ Then: $f$ preserves meet as a mapping from $\struct {\map {\it Ids} {\mathscr S}, \subseteq}$ into $\mathscr S$
Field Norm of Quaternion is Positive Definite
Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion. Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$. The field norm of $\mathbf x$: :$\map n {\mathbf x} := \cmod {\mathbf x \overline {\mathbf x} }$ is positive definite.
Field Norm of Quaternion is Multiplicative
Let $\mathbf x$ be a quaternion. Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$. The field norm of $\mathbf x$: :$\map n {\mathbf x} := \size {\mathbf x \overline {\mathbf x} }$ is a multiplicative function.
Equivalence of Definitions of Oscillation at Point for Real Functions
Let $X$ and $Y$ be real sets. Let $f: X \to Y$ be a real function. Let $x \in X$. {{TFAE|def = Oscillation/Real Space/Oscillation at Point|view = Oscillation at a Point}}
Supremum of Cartesian Product
Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ be an ordered set. Let $\left({S_1 \times S_2, \precsim}\right)$ be the Cartesian product of $\left({S_1, \preceq_1}\right)$ and $\left({S_2, \preceq_2}\right)$. Let $X_1$ be a non-empty subset of $S_1$, $X_2$ be a non-empty subset of $S_2$ such that :$X_1$ and $X_2$ admit suprema. Then :$X_1 \times X_2$ admits a supremum and :$\sup \left({X_1 \times X_2}\right) = \left({\sup X_1, \sup X_2}\right)$
Nonassociative Division Algebra with Real Scalars has Dimension of Power of 2
Let $A$ be a nonassociative division algebra with real scalars. Then the dimension of $A$ is a power of $2$.
Bott-Milnor-Kervaire 1,2,4,8 Theorem
Let $A$ be a division algebra with real scalars. Then the dimension of $A$ is either: :$1$: the real numbers $\R$ :$2$: the complex numbers $\C$ :$4$: the quaternions $\Bbb H$ or: :$8$: the octonions $\Bbb O$.
Montel's Theorem
Let $U \subseteq \C$ be an open subset of the complex numbers. Let $\mathcal H \left({U}\right)$ be the space of holomorphic mappings on $U$. Then a family of mappings $\mathcal F \subseteq \mathcal H \left({U}\right)$ is normal  $\mathcal F$ is locally bounded.
Vitali's Convergence Theorem
Let $U$ be an  open,  connected subset of $\C$. Let $S \subseteq U$ contain a limit point $\sigma$. Let $\sequence {f_n}_{n \mathop \in \N}$ be a  normal family of holomorphic mappings $f_n : U \to \C$. Let $\sequence {f_n}_{n \mathop \in \N}$  converge to some holomorphic mapping $f : U \to \C$ at $\sigma$. Then $f_n$ converges uniformly to $f$ on all compact subsets of $U$.
Identity Theorem
Let $U$ be an open connected subset of the complex plane $\C$. Let $f$ and $g$ be complex functions whose domain is $U$. Let $S = \left\{{z \in U: f \left({z}\right) = g \left({z}\right)}\right\}$. Let $f$ and $g$ be analytic on $U$. Let $S$ have a limit point in $U$. Then: :$\forall z \in U : f \left({z}\right) = g \left({z}\right)$
Exponential Sequence is Uniformly Convergent on Compact Sets
Let $\mathcal E = \left \langle{E_n}\right \rangle$ denote the sequence of functions $E_n: \C \to \C$ defined as: :$E_n \left({z}\right) = \left({1 + \dfrac z n}\right)^n$ Let $K$ be a compact subset of $\C$. Then $\mathcal E$ is uniformly convergent on $K$.
Uniformly Convergent Sequence Evaluated on Convergent Sequence
Let $X = \struct {A, d}$ and $Y = \struct {B, \rho}$ be metric spaces. {{explain|Definition:Complete Metric Space is invoked, but it is called just "metric space". Analyse to determine whether it needs to be complete here or not.}} Let $K$ be a subspace of $X$. Let $\FF = \sequence {f_n}$ be a sequence of continuous mappings $f_n: X \to Y$ uniformly convergent on $K$. Let $\sequence {a_n}$ be a Convergent Sequence in $K$ with limit $a \in K$. Then $\sequence {\map {f_n} {a_n} }$ is convergent such that: :$\displaystyle \lim_{n \mathop \to \infty} \map {f_n} {a_n} = \map f a$
Meet-Continuous iff Meet of Suprema equals Supremum of Meet of Ideals
Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be an up-complete lattice. Then :$\mathscr S$ is meet-continuous  :for every ideals $I, J$ in $\mathscr S$: $\paren {\sup I} \wedge \paren {\sup J} = \sup \set {i \wedge j: i \in I, j \in J}$
Ellipse is Mechanical Curve
The ellipse is a mechanical curve. That is, it cannot be constructed using a compass and straightedge. However, as many points as is desired can be so constructed that lie on any given ellipse.
Parabola is Mechanical Curve
The parabola is a mechanical curve. That is, it cannot be constructed using a compass and straightedge. However, as many points as is desired can be so constructed that lie on any given parabola.
Hyperbola is Mechanical Curve
The hyperbola is a mechanical curve. That is, it cannot be constructed using a compass and straightedge. However, as many points as is desired can be so constructed that lie on any given hyperbola.
Derivative of Sequence of Holomorphic Functions
Let $U$ be an  open,  connected subset of $\C$. Let $\left \langle{f_n}\right \rangle$ be a sequence of  holomorphic mappings $f_n: U \to \C$. Let $\left \langle{f_n}\right \rangle$  converge pointwise to some function $f: U \to \C$. Let $\left \langle{f_n}\right \rangle$  converge uniformly on  compact subsets of $U$. Then $f$ is holomorphic on $U$. Further, the sequence of  derivatives $\left \langle{f_n'}\right \rangle$ converges to $f'$ on $U$.  This convergence is uniform on compact subsets of $U$.
Uniformly Convergent Sequence of Bounded Functions is Uniformly Bounded
Let $X = \left({A, d}\right)$ and $Y = \left({B, \rho}\right)$ be metric spaces. Let $\left \langle{f_i}\right \rangle_{i \in I}$ be a uniformly convergent sequence of mappings $f_i: X \to Y$. $\forall i \in I$, let $f_i$ be bounded. Then $\left \langle{f_i}\right \rangle$ is uniformly bounded.
Product of Uniformly Convergent Sequences of Bounded Functions is Uniformly Convergent
Let $X = \struct {A, d}$ and $Y = \struct {B, \rho}$ be metric spaces. Let $\sequence {f_n}$ and $\sequence {g_n}$ be sequences of mappings from $X$ to $Y$. Let $\sequence {f_n}$ and $\sequence {g_n}$ be uniformly convergent on some subspace $S$ of $X$.  $\forall n \in \N$, let $f_n$ and $g_n$ be bounded. Then the sequence $\sequence {f_n g_n}$ is uniformly convergent on $S$. {{explain|for the product to be defined, we probably want $Y$ to have some more structure}}
Area between Radii and Curve in Polar Coordinates
Let $C$ be a curve expressed in polar coordinates $\polar {r, \theta}$ as: :$r = \map g \theta$ where $g$ is a real function. Let $\theta = \theta_a$ and $\theta = \theta_b$ be the two rays from the pole at angles $\theta_a$ and $\theta_b$ to the polar axis respectively. Then the area $\mathcal A$ between $\theta_a$, $\theta_b$ and $C$ is given by: :$\displaystyle \mathcal A = \int \limits_{\theta \mathop = \theta_a}^{\theta \mathop = \theta_b} \frac {\paren {\map g \theta}^2 \rd \theta} 2$ as long as $\paren {\map g \theta}^2$ is integrable.
Derivative of Exponential Function/Complex
The complex exponential function is its own derivative. That is: :$\map {D_z} {\exp z} = \exp z$
Exponential Sequence is Eventually Increasing
Let $\sequence {E_n}$ be the sequence of real functions $E_n: \R \to \R$ defined as: :$\map {E_n} x = \paren {1 + \dfrac x n}^n$ Then, for sufficiently large $n \in \N$, $\sequence {\map {E_n} x}$ is increasing with respect to $n$. That is: :$\forall x \in \R: \forall n \in \N: n \ge \ceiling {\size x} \implies \map {E_n} x \le \map {E_{n + 1} } x$ where $\ceiling x$ denotes the ceiling of $x$.
Euler's Number: Limit of Sequence implies Base of Logarithm
Let $e$ be  Euler's number defined by: :$\displaystyle e := \lim_{n \mathop \to \infty} \left({1 + \frac 1 n}\right) ^n$ Then $e$ is the unique solution to the equation $\ln \left({x}\right) = 1$. That is: :$\ln \left({x}\right) = 1 \iff x = e$
Diophantine Equation y cubed equals x squared plus 2
The indeterminate Diophantine equation: :$y^3 = x^2 + 2$ has only one integer solution: :$x = 5, y = 3$
Continuous Extension from Dense Subset
Let $X$ be a metric space. Let $D$ be a dense subset of $X$. Let $f: D \to \R$ be a uniformly continuous mapping. Then there exists a unique continuous extension of $f$ to $X$.
Sum of two Fourth Powers cannot be Fourth Power
$\forall a, b, c \in \Z_{>0}$, the equation $a^4 + b^4 = c^4$ has no solutions.
Moore-Osgood Theorem
Let $X$ and $Y$ be metric spaces. Let $S$ be a subspace of $X$. Let $c$ be a limit point of $S$. Let $\left \langle{f_n}\right \rangle$ be a sequence of mappings $f_n : X \to Y$. Suppose that: :$(1): \quad \left \langle{f_n}\right \rangle$ is uniformly convergent on $S$ :$(2): \quad \displaystyle \forall n \in \N : \lim_{x \to c} f_n \left({ x }\right)$ exists Then: :$\displaystyle \lim_{x \to c} \lim_{n \to \infty} f_n \left({ x }\right) = \lim_{n \to \infty} \lim_{x \to c} f_n \left({ x }\right)$
Cycloid is Best Shape for Arch
The optimal shape for an arch is a cycloid.
Center of Gravity of Cycloid
Let $C$ be a cycloid. The center of gravity of an arc of $C$ is at {{stub|research needed}}
Ore Graph is Connected
Let $G = \left({V, E}\right)$ be an Ore graph. Then $G$ is connected.
Power Function to Rational Power permits Unique Continuous Extension
Let $a \in \R_{> 0}$. Let $f: \Q \to \R$ be the real-valued function defined as: :$\map f q = a^q$ where $a^q$ denotes $a$ to the power of $q$. Then there exists a unique continuous extension of $f$ to $\R$.
Meet-Continuous iff Meet of Suprema equals Supremum of Meet of Directed Subsets
Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be an up-complete lattice. Then: :$\mathscr S$ is meet-continuous : :for every directed subsets $D_1, D_2$ of $S$: $\paren {\sup D_1} \wedge \paren {\sup D_2} = \sup \set {d_1 \wedge d_2: d_1 \in D_1, d_2 \in D_2}$
Brouwerian Lattice iff Meet-Continuous and Distributive
Let $\mathscr S = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Then :$\mathscr S$ is a Brouwerian lattice  :$\mathscr S$ is meet-continuous and distributive.
Power is Well-Defined/Integer
Let $x$ be a non-zero real number. Let $k$ be an integer. Then $x^k$ is well-defined.
Plane Figure with Maximum Area for given Perimeter is Circle
Let $F$ be a plane figure. Let $P$ be the length of the perimeter of $F$. Let the area of $F$ be the largest of all the plane figures whose perimeters are of length $P$. Then $F$ is a circle.
Real Number between Zero and One is Greater than Power/Natural Number
Let $x \in \R$. Let $0 < x < 1$. Let $n$ be a natural number. Then: : $0 < x^n \le x$
Power Function is Strictly Increasing over Positive Reals/Natural Exponent
Let $n \in \Z_{>0}$ be a strictly positive integer. Let $f: \R_{>0} \to \R$ be the real function defined as: :$f \left({x}\right) = x^n$ where $x^n$ denotes $x$ to the power of $n$. Then $f$ is strictly increasing.
Root is Strictly Increasing
Let $x \in \R_{> 0}$. Let $n \in \N$. Let $f: \R_{> 0} \to \R$ be the real function defined as: :$\map f x = \sqrt [n] x$ where $\sqrt [n] x$ denotes the $n$th root of $x$. Then $f$ is strictly increasing.
Shift Mapping is Lower Adjoint iff Meet is Distributive with Supremum
Let $\left({S, \preceq}\right)$ be a complete lattice. Then :$\forall x \in S, f: S \to S:\left({\forall y \in S: f\left({y}\right) = x \wedge y}\right) \implies f$ is lower adjoint of Galois connection  :$\forall x \in S, X \subseteq S: x \wedge \sup X = \sup \left\{ {x \wedge y: y \in X}\right\}$
Greatest Common Divisor is Associative
Let $a, b, c \in \Z$. Then: :$\gcd \left\{ {a, \gcd \left\{{b, c}\right\} }\right\} = \gcd \left\{ {\gcd \left\{{a, b}\right\}, c}\right\}$ where $\gcd$ denotes the greatest common divisor.
Lowest Common Multiple is Associative
Let $a, b, c \in \Z$. Then: :$\lcm \left\{{a, \lcm \left\{{ b , c }\right\} }\right\} = \lcm \left\{{\lcm \left\{{a, b}\right\}, c}\right\}$ where $\lcm$ denotes the lowest common multiple.
LCM of Coprime Integers
Let $a, b \in \Z_{>0}$ be coprime integers. Then: :$\lcm \set {a, b} = a b$ where $\lcm$ denotes the lowest common multiple.
GCD with One Fixed Argument is Multiplicative Function
Let $a, b, c \in \Z: b \perp c$ where $b \perp c$ denotes that $b$ is coprime to $c$. Then: :$\gcd \set {a, b} \gcd \set {a, c} = \gcd \set {a, b c}$ That is, GCD is multiplicative.
Power Function on Base between Zero and One is Strictly Decreasing/Positive Integer
Let $a \in \R$ be a real number such that $0 < a < 1$. Let $f: \Z_{\ge 0} \to \R$ be the real-valued function defined as: :$\map f n = a^n$ where $a^n$ denotes $a$ to the power of $n$. Then $f$ is strictly decreasing.
Power Function on Base between Zero and One is Strictly Decreasing/Rational Number
Let $a \in \R$ be a real number such that $0 < a < 1$. Let $f: \Q \to \R$ be the real-valued function defined as: :$\map f q = a^q$ where $a^q$ denotes $a$ to the power of $q$. Then $f$ is strictly decreasing.
Power Function on Base Greater than One is Strictly Increasing/Positive Integer
Let $a \in \R$ be a real number such that $a > 1$. Let $f: \Z_{\ge 0} \to \R$ be the real-valued function defined as: :$\map f n = a^n$ where $a^n$ denotes $a$ to the power of $n$. Then $f$ is strictly increasing.
Root of Reciprocal is Reciprocal of Root
Let $x \in \R_{\ge 0}$. Let $n \in \N$. Let $\sqrt [n] x$ denote the $n$th root of $x$. Then: :$\sqrt [n] {\dfrac 1 x} = \dfrac 1 {\sqrt [n] x}$
Power Function on Base Greater than One is Strictly Increasing/Rational Number
Let $a \in \R$ be a real number such that $a > 1$. Let $f: \Q \to \R$ be the real-valued function defined as: :$\map f q = a^q$ where $a^q$ denotes $a$ to the power of $q$. Then $f$ is strictly increasing.
Existence of Square Roots of Positive Real Number
Let $r \in \R_{\ge 0}$ be a positive real number. Then: :$\exists y_1 \in \R_{\ge 0}: {y_1}^2 = r$ :$\exists y_2 \in \R_{\le 0}: {y_2}^2 = r$
Inequality iff Difference is Positive
Let $x, y \in \R$. Then the following are equivalent: :$(1): \quad x < y$ :$(2): \quad y - x > 0$
Power Function on Base between Zero and One is Strictly Decreasing/Integer
Let $a \in \R$ be a real number such that $0 < a < 1$. Let $f: \Z \to \R$ be the real-valued function defined as: :$\map f k = a^k$ where $a^k$ denotes $a$ to the power of $k$. Then $f$ is strictly decreasing.
Meet-Continuous iff Meet Preserves Directed Suprema
Let $\mathscr S = \left({S, \vee, \wedge, \preceq}\right)$ be an up-complete lattice. Let $\left({S \times S, \precsim}\right)$ be the Cartesian product of $\mathscr S$ and $\mathscr S$. Let $f: S \times S \to S$ be a mapping such that :$\forall x, y \in S: f\left({x, y}\right) = x \wedge y$ Then :$\mathscr S$ is meet-continuous  :$f$ preserves directed suprema
Real Star-Algebra is Commutative
Let $A = \left( A_F , \oplus \right)$ be a real *-algebra whose conjugation is denoted as $*$. Then: :$\forall a,b \in A, a \oplus b = b \oplus a$ That is, real *-algebra is commutative.
Meet-Continuous implies Shift Mapping Preserves Directed Suprema
Let $\mathscr S = \left({S, \vee, \wedge, \preceq}\right)$ be a meet-continuous lattice. Let $x \in S$. Let $f: S \to S$ be a mapping such that :$\forall y \in S: f\left({y}\right) = x \wedge y$ Then :$f$ preserves directed suprema.
Power of Strictly Positive Real Number is Strictly Positive/Positive Integer
Let $x \in \R_{>0}$ be a (strictly) positive real number. Let $n \in \Z_{\ge 0}$ be a positive integer. Then: :$x^n > 0$ where $x^n$ denotes the $n$th power of $x$.
Power of Positive Real Number is Positive/Integer
Let $x \in \R_{>0}$ be a (strictly) positive real number. Let $n \in \Z$ be an integer. Then: :$x^n > 0$ where $x^n$ denotes the $n$th power of $x$.
Power of Positive Real Number is Positive/Rational Number
Let $x \in \R_{>0}$ be a (strictly) positive real number. Let $q \in \Q$ be a rational number. Then: :$x^q > 0$ where $x^q$ denotes the $x$ to the power of $q$.
Power Function is Monotone/Rational Number
Let $a \in \R_{>0}$. Let $f: \Q \to \R$ be the real-valued function defined as: :$\map f r = a^r$ where $a^r$ denotes $a$ to the power of $r$. Then $f$ is monotone. Further, $f$ is strictly monotone unless $a = 1$.
Power Function tends to One as Power tends to Zero/Rational Number
Let $a \in \R_{> 0}$. Let $f: \Q \to \R$ be the real-valued function defined as: :$\map f q = a^q$ where $a^q$ denotes $a$ to the power of $q$. Then: :$\displaystyle \lim_{x \mathop \to 0} \map f x = 1$
Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number/Lemma
Let $a \in \R$ be a real number such that $a > 1$. Let $r \in \Q_{> 0}$ be a strictly positive rational number such that $r < 1$. Then: :$1 < a^r < 1 + a r$
Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number
Let $a \in \R_{> 0}$ be a strictly positive real number such that $a > 1$. Let $f: \Q \to \R$ be the real-valued function defined as: :$\map f r = a^r$ where $a^r$ denotes $a$ to the power of $r$. Then: :$\displaystyle \lim_{r \mathop \to 0} \map f r = 1$
Power Function on Base between Zero and One Tends to One as Power Tends to Zero/Rational Number
Let $a \in \R_{> 0}$ be a strictly positive real number such that $0 < a < 1$. Let $f: \Q \to \R$ be the real-valued function defined as: :$\map f r = a^r$ where $a^r$ denotes $a$ to the power of $r$. Then: :$\displaystyle \lim_{r \mathop \to 0} \map f r = 1$
Euler's Number to Rational Power permits Unique Continuous Extension
Let $e$ be Euler's number. Let $f: \Q \to \R$ be the real-valued function defined as: :$f \left({q}\right) = e^q$ where $e^q$ denotes  $e$ to the power of $q$. Then there exists a unique continuous extension of $f$ to $\R$.
Open Unit Interval is Proper Subset of Closed Unit Interval
The open unit interval: :$I_o = \left({0 \,.\,.\, 1}\right)$ is a proper subset of the closed unit interval: :$I_c = \left[{0 \,.\,.\, 1}\right]$
Vector Cross Product satisfies Jacobi Identity
Let $\mathbf a, \mathbf b, \mathbf c$ be vectors in $3$ dimensional Euclidean space. Let $\times$ denotes the cross product. Then: :$\mathbf a \times \paren {\mathbf b \times \mathbf c} + \mathbf b \times \paren {\mathbf c \times \mathbf a} + \mathbf c \times \paren {\mathbf a \times \mathbf b} = \mathbf 0$ That is, the cross product operation satisfies the Jacobi identity.
Meet-Continuous and Distributive implies Shift Mapping Preserves Finite Suprema
Let $\left({S, \vee, \wedge, \preceq}\right)$ be a meet-continuous distributive complete lattice. Let $x \in S$. Let $f: S \to S$ be a mapping such that :$\forall y \in S: f \left({y}\right) = x \wedge y$ Then :$f$ preserves finite suprema
Set Union is not Cancellable
Set union is not a cancellable operation. That is, for a given $A, B, C \subseteq S$ for some $S$, it is not always the case that: :$A \cup B = A \cup C \implies B = C$
Intersection of Real Intervals is Real Interval
Let $I_1$ and $I_2$ be real intervals. Then $I_1 \cap I_2$ is also a real interval.
Union of Real Intervals is not necessarily Real Interval
Let $I_1$ and $I_2$ be real intervals. Then $I_1 \cup I_2$ is not necessarily a real interval.
Pasting Lemma/Corollary 1
Let $A$ and $B$ be closed in $X$. Let $f : A \to Y$ and $g : B \to Y$ be continuous mappings that agree on $A \cap B$. Then the mapping $f \cup g : A \cup B \to Y$ is continuous.
Pasting Lemma/Corollary 2
Let $\mathcal A = \left\{ {A_i: i \in I} \right\}$ be a set of sets that are open in $X$. Let $f: \bigcup \mathcal A \to Y$ be a mapping such that: :$\forall i \in I : f \restriction A_i$ is continuous Then $f$ is continuous on $\bigcup \mathcal A$.
Pasting Lemma
Let $X$ and $Y$ be topological spaces. Let $A$ and $B$ be open in $X$. Let $f: A \to Y$ and $g: B \to Y$ be continuous mappings that agree on $A \cap B$. Let $f \cup g$ be the union of the mappings $f$ and $g$: :$\forall x \in A \cup B: \map {f \cup g} x = \begin {cases} \map f x & : x \in A \\ \map g x & : x \in B \end {cases}$ Then the mapping $f \cup g : A \cup B \to Y$ is continuous.
Domain of Real Square Root Function
The domain of the real square root function is the set of positive real numbers $\R_{\ge 0}$: :$\left\{ {x \in \R: x \ge 0}\right\}$
Image of Real Square Root Function
The image of the real square root function is the entire set of positive real numbers $\R_{\ge 0}$.
Exponent Combination Laws/Product of Powers/Proof 1
Let $a \in \R_{> 0}$ be a positive real number. Let $x, y \in \R$ be real numbers. Let $a^x$ be defined as $a$ to the power of $x$. Then: :$a^x a^y = a^{x + y}$
Inequality of Product of Unequal Numbers
Let $a, b, c, d \in \R$. Then: :$0 < a < b \land 0 < c < d \implies 0 < a c < b d$
Negative of Absolute Value/Corollary 3
Let $y \in \R_{\ge 0}$. Let $z \in \R$. Then: :$\left\vert{x - z}\right\vert < y \iff z - y < x < z + y$
Exponent Combination Laws/Product of Powers/Proof 2/Lemma
Let $x_1, x_2, y_1, y_2 \in \R_{>0}$ be strictly positive real numbers. Let $\epsilon \in \openint 0 {\min \set {y_1, y_2, 1} }$. Then: :$\size {x_1 - y_1} < \epsilon \land \size {x_2 - y_2} < \epsilon \implies \size {x_1 x_2 - y_1 y_2} < \epsilon \paren {y_1 + y_2 + 1}$
Union of Unordered Tuples
Let $x_1, \dots, x_n, x_{n+1}, \dots, x_m$ be arbitrary. Then :$\set {x_1, \dots, x_n} \cup \set {x_{n + 1}, \dots, x_m} = \set {x_1, \dots, x_n, x_{n + 1}, \dots, x_m}$
Gauss-Lucas Theorem
Let $P$ be a (nonconstant) polynomial with complex coefficients. Then all zeros of its derivative $P'$  belong to the convex hull of the set of zeros of $P$.
Way Below implies Preceding
Let $\left({S, \preceq}\right)$ be an ordered set. Let $x, y \in S$ such that :$x \ll y$ where $\ll$ denotes element is way below second element. Then :$x \preceq y$
Preceding and Way Below implies Way Below
Let $\left({S, \preceq}\right)$ be an ordered set. Let $u, x, y, z \in S$ such that :$u \preceq x \ll y \preceq z$ where $\ll$ denotes the way below relation. Then :$u \ll z$
Join is Way Below if Operands are Way Below
Let $\left({S, vee, \preceq}\right)$ be a join semilattice. Let $x, y, z \in S$ such that :$x \ll z$ and $y \ll z$ where $\ll$ denotes the way below relation. Then :$x \vee y \ll z$
Way Below Relation is Transitive
Let $\left({S, \preceq}\right)$ be an ordered set. Let $x, y, z \in S$ such that :$x \ll y \ll z$ Then :$x \ll z$
Way Below Relation is Antisymmetric
Let $\left({S, \preceq}\right)$ be an ordered set. Let $x, y \in S$ such that :$x \ll y$ and $y \ll x$ Then :$x = y$
Continued Fraction Expansion of Golden Mean
The golden mean has the simplest possible continued fraction expansion, namely $\sqbrk {1, 1, 1, 1, \ldots}$: :$\phi = 1 + \cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots} } }$