hcju/proofwikips huggingface dataset:
Fibonacci Number less than Golden Section to Power less One
For all $n \in \N_{> 0}$: :$F_n \le \phi^{n - 1}$ where: :$F_n$ is the $n$th Fibonacci number :$\phi$ is the golden section: $\phi = \dfrac {1 + \sqrt 5} 2$
Square of Golden Mean equals One plus Golden Mean
:$\phi^2 = \phi + 1$ where $\phi$ denotes the golden mean.
Fibonacci Number greater than Golden Section to Power less Two
For all $n \in \N_{\ge 2}$: :$F_n \ge \phi^{n - 2}$ where: :$F_n$ is the $n$th Fibonacci number :$\phi$ is the golden section: $\phi = \dfrac {1 + \sqrt 5} 2$
Triangular Number as Alternating Sum and Difference of Squares
{{begin-eqn}} {{eqn | ll= \forall n \in \N:        | l = \frac {n \paren {n + 1} } 2       | r = \sum_{j \mathop = 0}^{n - 1} \paren {-1}^j \paren {n - j}^2       | c =  }} {{eqn | r = n^2 - \paren {n - 1}^2 + \paren {n - 2}^2 - \cdots + \paren {-1}^{n - 1}        | c =  }} {{end-eqn}} Thus the $n$th triangular number can be expressed as the alternating sum and difference of squares: So: {{begin-eqn}} {{eqn | l = 1       | r = 1^2       | c =  }} {{eqn | l = 3       | r = 2^2 - 1^2       | c =  }} {{eqn | l = 6       | r = 3^2 - 2^2 + 1^2       | c =  }} {{eqn | l = 10       | r = 4^2 - 3^2 + 2^2 - 1^2       | c =  }} {{end-eqn}} and so on.
Bernoulli's Inequality/Corollary
Let $x \in \R$ be a real number such that $0 < x < 1$. Let $n \in \Z_{\ge 0}$ be a positive integer. Then: :$\left({1 - x}\right)^n \ge 1 - n x$
2 to the n is Greater than n Cubed when n is 10 and above
:$\forall n \in \Z, n \ge 10: 2^n > n^3$
Set of Integers can be Well-Ordered
The set of integers $\Z$ can be well-ordered with an appropriately chosen ordering.
Existence of Digital Root
Let $n \in \N$ be a natural number. Let $b \in \N$ such that $b \ge 2$ also be a natural number. Let $n$ be expressed in base $b$. Then the digital root base $b$ exists for $n$.
Golden Mean is Irrational
The golden mean $\phi$ is irrational.
Way Below iff Preceding Finite Supremum
Let $\left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $x, y \in S$. Then $x \ll y$  :$\forall X \subseteq S: y \preceq \sup X \implies \exists A \in {\it Fin}\left({X}\right): x \preceq \sup A$ where :$\ll$ denotes the way below relation, :${\it Fin}\left({X}\right)$ denotes the set of all finite subsets of $X$.
Exponent Combination Laws/Product of Powers/Proof 2
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: : $a^x a^y = a^{x + y}$
Exponent Combination Laws/Power of Power/Proof 1
Let $a \in \R_{>0}$ be a (strictly) positive real number. {{:Exponent Combination Laws/Power of Power}}
Exponent Combination Laws/Power of Power/Proof 2
Let $a \in \R_{>0}$ be a (strictly) positive real number. {{:Exponent Combination Laws/Power of Power}}
Rational Sequence Increasing to Real Number
Let $x \in \R$ be a real number. Then there exists some increasing rational sequence that converges to $x$.
Rational Sequence Decreasing to Real Number
Let $x \in \R$ be a real number. Then there exists some decreasing rational sequence that converges to $x$.
Power Function on Base Greater than One is Strictly Increasing/Real Number
Let $a \in \R$ be a real number such that $a > 1$. Let $f: \R \to \R$ be the real function defined as: :$\map f x = a^x$ where $a^x$ denotes $a$ to the power of $x$. Then $f$ is strictly increasing.
Power Function on Strictly Positive Base is Continuous/Real Power
Let $f : \R \to \R$ be the real function defined as: :$\map f x = a^x$ where $a^x$ denotes $a$ to the power of $x$. Then $f$ is continuous.
Power Function on Strictly Positive Base is Continuous/Rational Power
Let $f: \Q \to \R$ be the real-valued function defined as: :$\map f x = a^x$ where $a^x$ denotes $a$ to the power of $x$. Then $f$ is continuous.
Defining Sequence of Natural Logarithm is Convergent
Let $x \in \R$ be a real number such that $x > 0$. Let $\left\langle{ f_n }\right\rangle$ be the sequence of mappings $f_n : \R_{>0} \to \R$ defined as: :$f_n \left({ x }\right) = n \left({ \sqrt[n]{ x } - 1 }\right)$ Then $\left\langle{ f_n }\right\rangle$ is pointwise convergent.
Exponent Combination Laws/Positive Integers
Let $r \in \R_{>0}$ be (strictly) positive real numbers.
Sum of Indices of Real Number/Rational Numbers
Let $x, y \in \Q$ be rational numbers. Let $r^x$ be defined as $r$ to the power of $n$. Then: : $r^{x + y} = r^x \times r^y$
Exponent Combination Laws/Rational Numbers
Let $r \in \R_{> 0}$ be a (strictly) positive real number.
Product of Indices of Real Number/Rational Numbers
Let $x, y \in \Q$ be rational numbers. Let $r^x$ be defined as $r$ to the power of $x$. Then: :$\paren {r^x}^y = r^{x y}$
Sum of Indices of Real Number
Let $r \in \R_{> 0}$ be a (strictly) positive real number.
Product of Indices of Real Number
Let $r \in \R_{> 0}$ be a (strictly) positive real number.
Sum of Indices of Real Number/Integers
Let $n, m \in \Z$ be integers. Let $r^n$ be defined as $r$ to the power of $n$. Then: :$r^{n + m} = r^n \times r^m$
Product of Indices of Real Number/Integers
Let $n, m \in \Z$ be positive integers. Let $r^n$ be defined as $r$ to the power of $n$. Then: :$\paren {r^n}^m = r^{n m}$
Uniformly Convergent iff Difference Under Supremum Metric Vanishes
Let $X$ and $Y$ be metric spaces Let $\sequence {f_n}$ be a sequence of mappings defined on $X$. Let $f: X \to Y$ be a mapping. Let $d_S: S \times S \to Y$ denote the supremum metric on $S \subseteq X$. Then $\sequence {f_n}$ converges uniformly to $f$ on $S$  $\map {d_S} {f_n, f} \to 0$ as $n \to \infty$.
Uniform Convergence is Hereditary
Let $M = \struct {A, d}$ be a metric space. Let $\sequence {f_n}$ be a sequence of mappings defined on $A$. Let $\sequence {f_n}$ be uniformly convergent on $S \subseteq A$. Then $\sequence {f_n}$ is uniformly convergent on every metric subspace of $S$. That is, uniform convergence is a hereditary property of a metric space.
Real Number to Negative Power
Let $r \in \R_{> 0}$ be a (strictly) positive real number.
Real Number to Negative Power/Integer
Let $n \in \Z$ be an integer. Let $r^n$ be defined as $r$ to the power of $n$. Then: :$r^{-n} = \dfrac 1 {r^n}$
Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand is Element of Ideal
Let $\mathscr S = \left({S, \preceq}\right)$ be an up-complete ordered set. Let $x, y \in S$. Then $x \ll y$  :$\forall I \in {\it Ids}\left({\mathscr S}\right): y \preceq \sup I \implies x \in I$ where :$\ll$ denotes the way below relation, :${\it Ids}\left({\mathscr S}\right)$ denotes the set of all ideals in $\mathscr S$.
Lower Closure of Directed Subset is Ideal
Let $\mathscr S = \left({S, \preceq}\right)$ be an ordered set. Let $D$ be a directed subset of $S$. Then :$D^\preceq$ is an ideal in $\mathscr S$ where $D^\preceq$ denotes the lower closure of $D$.
Sum of Geometric Sequence/Corollary 2
:$\displaystyle \sum_{j \mathop = 0}^{n - 1} j x^j = \frac {\paren {n - 1} x^{n + 1} - n x^n + x} {\paren {x - 1}^2}$
Defining Sequence of Natural Logarithm is Strictly Decreasing
Let $x \in \R$ be a real number such that $x > 0$. Let $\sequence {f_n}$ be the sequence of mappings $f_n : \R_{>0} \to \R$ defined as: :$\map {f_n} x = n \paren {\sqrt [n] x - 1}$ Then $\forall x \in \R_{>0}: \sequence {\map {f_n} x}$ is strictly decreasing.
Logarithm Base 10 of 2 is Irrational
The common logarithm of $2$: :$\log_{10} 2 \approx 0.30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots$ is irrational.
Nth Root of 1 plus x not greater than 1 plus x over n
Let $x \in \R_{>0}$ be a (strictly) positive real number. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: :$\sqrt [n] {1 + x} \le 1 + \dfrac x n$
Reciprocal of Logarithm
Let $x, y \in \R_{> 0}$ be (strictly) positive real numbers. Then: :$\dfrac 1 {\log_x y} = \log_y x$
Discontinuity of Monotonic Function is Jump Discontinuity
Let $X$ be an open subset of $\R$. Let $f: X \to Y$ be a monotone real function. Then $f$ is discontinuous at a point $c \in X$  $c$ is a jump discontinuity of $f$.
Surjective Monotone Function is Continuous
Let $X$ be an open subset of $\R$. Let $Y$ be a real interval. Let $f: X \to Y$ be a surjective monotone real function. Then $f$ is continuous on $X$.
Limit of Bounded Convergent Sequence is Bounded
Let $\left \langle {x_n} \right \rangle$, $\left \langle {a_n} \right \rangle$, and $\left \langle {b_n} \right \rangle$ be convergent sequences in $\R$. Let $\left \langle {x_n} \right \rangle$, $\left \langle {a_n} \right \rangle$, and $\left \langle {b_n} \right \rangle$ converge to $x, a, b \in \R$, respectively. Suppose that: :$\exists N \in \N : n \geq N \implies a_n \leq x_n \leq b_n$ Then: :$ a \le x \le b$
Logarithm of Power/Natural Logarithm/Natural Power
Let $x \in \R$ be a strictly positive real number. Let $n \in \R$ be any natural number. Let $\ln x$ be the natural logarithm of $x$. Then: :$\ln \left({x^n}\right) = n \ln x$
Logarithm of Power/Natural Logarithm/Integer Power
Let $x \in \R$ be a strictly positive real number. Let $n \in \R$ be any integer. Let $\ln x$ be the natural logarithm of $x$. Then: :$\ln \left({ x^n }\right) = n \ln x$
Logarithm of Power/Natural Logarithm/Rational Power
Let $x \in \R$ be a strictly positive real number. Let $r \in \R$ be any rational number. Let $\ln x$ be the natural logarithm of $x$. Then: :$\ln \left({ x^r }\right) = r \ln x$
Dini's Theorem
Let $K \subseteq \R$ be compact. Let $\sequence {f_n}$ be a sequence of continuous real functions defined on $K$. Let $\sequence {f_n}$ converge pointwise to a continuous function $f$. Suppose that: :$\forall x \in K : \sequence {\map {f_n} x}$ is monotone. Then the convergence of $\sequence {f_n}$ to $f$ is uniform.
Way Below in Meet-Continuous Lattice
Let $\mathscr S = \left({S, \vee, \wedge, \preceq}\right)$ be a meet-continuous bounded below lattice. Let $x, y \in S$. Then $x \ll y$  :$\forall I \in {\it Ids}\left({\mathscr S}\right): y = \sup I \implies x \in I$ where :$\ll$ denotes the way below relation, :${\it Ids}\left({\mathscr S}\right)$ denotes the set of all ideals in $\mathscr S$.
Approximation to Binary Logarithm from Natural and Common Logarithm
The binary logarithm $\lg x$ can be approximated, to within $1 \%$, by the expression: :$\lg x \approx \ln x + \log_{10} x$ That is, by the sum of the natural logarithm and common logarithm.
Monotone Real Function with Everywhere Dense Image is Continuous
Let $I$ and $J$ be real intervals. Let $f: I \to J$ be a monotone real function. Let $f \sqbrk I$ be everywhere dense in $J$, where $f \sqbrk I$ denotes the image of $I$ under $f$. Then $f$ is continuous on $I$.
Relative Difference between Infinite Set and Finite Set is Infinite
Let $S$ be an infinite set. Let $T$ be a finite set. Then $S \setminus T$ is an infinite set.
Set Difference over Subset
Let $A$, $B$, and $S$ be sets. Let $A \subseteq B$. Then: :$A \setminus S \subseteq B \setminus S$
Subset of Empty Set iff Empty
Let $S$ be a set. Let $\O$ denote the empty set. Then $S \subseteq \O$  $S = \O$.
Finite Subset of Metric Space is Closed
Let $M = \struct {A, d}$ be a metric space. Let $S \subseteq A$ be finite. Then $S$ is closed in $M$.
Monotone Real Function with Everywhere Dense Image is Continuous/Lemma
:$\displaystyle \openint {\lim_{x \mathop \to c^-} \map f x} {\lim_{x \mathop \to c+ } \map f x} \cap f \sqbrk I \subseteq \set {\map f c}$
Defining Sequence of Natural Logarithm is Uniformly Convergent on Compact Sets
Let $x \in \R$ be a real number such that $x > 0$. Let $\sequence {f_n}$ be the sequence of mappings $f_n : \R_{>0} \to \R$ defined as: :$\map {f_n} x = n \paren {\sqrt [n] x - 1}$ Let $K \subseteq \R_{>0}$ be compact. Then $\sequence {f_n}$ is uniformly convergent on $K$.
Multiplication of Positive Number by Real Number Greater than One
Let $x$ and $y$ be real numbers. Let $x > 1$. Let $y > 0$. Then $\dfrac y x < y$.
Set of Finite Suprema is Directed
Let $\left({S, \vee, \preceq}\right)$ be a join semilattice. Let $X$ be a non-empty subset of $S$. Then :$\left\{ {\sup A: A \in {\it Fin}\left({X}\right) \land A \ne \varnothing}\right\}$ is directed. where ${\it Fin}\left({X}\right)$ denotes the set of all finite subsets of $X$.
Change of Index Variable of Summation
:$\displaystyle \sum_{R \left({i}\right)} a_i = \sum_{R \left({j}\right)} a_j$
Singleton is Finite
Let $x$ be arbitrary. Then $\set x$ is a finite set.
T4 Property Preserved in Closed Subspace/Corollary
If $T$ is a normal space then $T_K$ is also a normal space. That is, the property of being a normal space is weakly hereditary.
Derivative of Uniformly Convergent Sequence of Differentiable Functions
Let $J$ be a bounded interval. Let $\left\langle{f_n}\right\rangle$ be a sequence of real functions $f_n: J \to \R$. Let each of $\left\langle{f_n}\right\rangle$ be differentiable on $J$. Let $\left\langle{ f_n \left({ x_0 }\right) }\right\rangle$ be convergent for some $x_0 \in J$. Let the sequence of derivatives $\left\langle{f_n'}\right\rangle$ converge uniformly on $J$ to a function $g : J \to \R$. Then $\left\langle{f_n}\right\rangle$ converge uniformly on $J$ to a differentiable function $f: J \to \R$ and $D_x f = g$.
Derivative of Natural Logarithm Function/Proof 4/Lemma
Let $\sequence {f_n}_n$ be the sequence of real functions $f_n: \R_{>0} \to \R$ defined as: :$\map {f_n} x = n \paren {\sqrt [n] x - 1}$ Let $k \in \N$. Let $J = \closedint {\dfrac 1 k} k$. Then the sequence of derivatives $\sequence { {f_n}'}_n$ converges uniformly to some real function $g: J \to \R$.
Exponential Function is Well-Defined/Real
Let $x \in \R$ be a real number. Let $\exp x$ be the exponential of $x$. Then $\exp x$ is well-defined.
Tail of Convergent Sequence
Let $\left\langle{a_n}\right\rangle$ be a real sequence. Let $N \in \N$ be a natural number. Let $a \in R$ be a real number. Then: :$a_n \to a$ : :$a_{n + N} \to a$
Union of Inverses of Mappings is Inverse of Union of Mappings
Let $I$ be an indexing set. Let $\family {f_i: i \in I}$ be an indexed family of mappings. For each $i \in I$, let $f^{-1}$ denote the inverse of $f$. Then the inverse of the union of $\family {f_i: i \in I}$ is the union of the inverses of $f_i, i \in I$. That is: :$\ds \paren {\bigcup \family {f_i: i \in I} }^{-1} = \bigcup \family {f_i^{-1}: i \in I}$
Union of Functions Theorem/Corollary
For each $i \in \N$, let $g_i : X_i \to Y$ be invertible. Then $\displaystyle \bigcup \left\{{g_i: i \in \N}\right\}$ is invertible and: :$\displaystyle \left({\bigcup \left\{{g_i: i \in \N}\right\} }\right)^{-1} = \bigcup \left\{{g_i^{-1}: i \in \N}\right\}$
Permutation of Indices of Summation
:$\displaystyle \sum_{\map R j} a_j = \sum_{\map R {\map \pi j} } a_{\map \pi j}$
Translation of Index Variable of Summation
Let $R: \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ be a propositional function on the set of integers. Let $\displaystyle \sum_{R \left({j}\right)} a_j$ denote a summation over $R$. Then: :$\displaystyle \sum_{R \left({j}\right)} a_j = \sum_{R \left({c \mathop + j}\right)} a_{c \mathop + j} = \sum_{R \left({c \mathop - j}\right)} a_{c \mathop - j}$ where $c$ is an integer constant which is not dependent upon $j$.
Axiom of Approximation in Up-Complete Semilattice
Let $\mathscr S = \left({S, \wedge, \preceq}\right)$ be an up-complete meet semilattice. Let :$\forall x \in S: x^\ll$ is directed Then :$\mathscr S$ satisfies axiom of approximation  :$\forall x, y \in S: x \npreceq y \implies \exists u \in S: u \ll x \land u \npreceq y$
Operand is Upper Bound of Way Below Closure
Let $\struct {S, \preceq}$ be an ordered set. Let $x \in S$. Then :$x$ is upper bound for $x^\ll$ where $x^\ll$ denotes the way below closure of $x$.
Exchange of Order of Summation
:$\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({j}\right)} a_{i j} = \sum_{S \left({j}\right)} \sum_{R \left({i}\right)} a_{i j}$
Sum of Summations equals Summation of Sum/Infinite Sequence
Let $R: \Z \to \set {\mathrm T, \mathrm F}$ be a propositional function on the set of integers $\Z$. Let $\displaystyle \sum_{\map R i} x_i$ denote a summation over $R$. Let the fiber of truth of $R$ be infinite. Let $\displaystyle \sum_{\map R i} b_i$ and $\displaystyle \sum_{\map R i} c_i$ be convergent. Then: :$\displaystyle \sum_{\map R i} \paren {b_i + c_i} = \sum_{\map R i} b_i + \sum_{\map R i} c_i$
Exchange of Order of Summation with Dependency on Both Indices
:$\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({i, j}\right)} a_{i j} = \sum_{S' \left({j}\right)} \sum_{R' \left({i, j}\right)} a_{i j}$ where: :$S' \left({j}\right)$ denotes the propositional function: ::there exists an $i$ such that both $R \left({i}\right)$ and $S \left({i, j}\right)$ hold :$R' \left({i, j}\right)$ denotes the propositional function: ::both $R \left({i}\right)$ and $S \left({i, j}\right)$ hold.
Summation of i from 1 to n of Summation of j from 1 to i
:$\displaystyle \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^i a_{i j} = \sum_{j \mathop = 1}^n \sum_{i \mathop = j}^n a_{i j}$
Exchange of Order of Summation/Infinite Series
Let the fiber of truth of both $R$ and $S$ be infinite. Let: :$\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({j}\right)} \left\vert{a_{i j} }\right\vert$ exist. Then: :$\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({j}\right)} a_{i j} = \sum_{S \left({j}\right)} \sum_{R \left({i}\right)} a_{i j}$
Exchange of Order of Summation with Dependency on Both Indices/Infinite Series
Let the fiber of truth of both $R$ and $S$ be infinite. Let: :$\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({i, j}\right)} \left\vert{a_{i j} }\right\vert$ exist. Then: :$\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({i, j}\right)} a_{i j} = \sum_{S' \left({j}\right)} \sum_{R' \left({i, j}\right)} a_{i j}$ where: :$S' \left({j}\right)$ denotes the propositional function: ::there exists an $i$ such that both $R \left({i}\right)$ and $S \left({i, j}\right)$ hold :$R' \left({i, j}\right)$ denotes the propositional function: ::both $R \left({i}\right)$ and $S \left({i, j}\right)$ hold.
Exchange of Order of Summation/Finite and Infinite Series
Let the fiber of truth of $R$ be infinite. Let the fiber of truth of $S$ be finite. For all $j$ in the fiber of truth of $S$, let $\displaystyle \sum_{R \left({i}\right)} a_{i j}$ be convergent. Then: :$\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({j}\right)} a_{i j} = \sum_{S \left({j}\right)} \sum_{R \left({i}\right)} a_{i j}$
Permutation of Indices of Summation/Infinite Series
Let the fiber of truth of $R$ be infinite. Let $\displaystyle \sum_{\map R i} a_i$ be absolutely convergent. Then: :$\displaystyle \sum_{\map R j} a_j = \sum_{\map R {\map \pi j} } a_{\map \pi j}$
Sum of Summations equals Summation of Sum
Let $R: \Z \to \set {\T, \F}$ be a propositional function on the set of integers. Let $\displaystyle \sum_{\map R i} x_i$ denote a summation over $R$. Let the fiber of truth of $R$ be finite. Then: :$\displaystyle \sum_{\map R i} \paren {b_i + c_i} = \sum_{\map R i} b_i + \sum_{\map R i} c_i$
Sum of Summations over Overlapping Domains
:$\displaystyle \sum_{R \left({j}\right)} a_j + \sum_{S \left({j}\right)} a_j = \sum_{R \left({j}\right) \mathop \lor S \left({j}\right)} a_j + \sum_{R \left({j}\right) \mathop \land S \left({j}\right)} a_j$ where $\lor$ and $\land$ signify logical disjunction and logical conjunction respectively.
Rules for Manipulating Summations
Let $R: \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ and $S: \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ be propositional functions on the set of integers. Let $S: \Z \times \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ be a propositional functions on the Cartesian product of the set of integers with itself. Let $\displaystyle \sum_{R \left({i}\right)} x_i$ denote a summation over $R$. Let $\pi$ be a permutation on the fiber of truth of $R$.
Uniqueness of Continuously Differentiable Solution to Initial Value Problem
Let $D \subseteq \R^2$ be a region containing $\tuple {a, b}$. Let $f: D \to \R$ be real-valued mapping such that $f$ and $\dfrac {\partial f} {\partial x}$ are continuous on $D$. Consider the initial value problem: :$\dfrac {\d x} {\d t} = \map f {t, x}$ :$\map x a = b$ Suppose the initial value problem above has a solution $\phi$ for all $x$ in some interval $J$ containing $a$. Then the solution $\phi$ is  unique on $J$.
Exhausting Sequence of Sets on the Strictly Positive Real Numbers
For each $k \in \N$, let $S_k = \openint {\dfrac 1 k} k$. Then $\sequence {S_k}_k$ is an exhausting sequence of sets on $\R_{>0}$.
Way Below in Ordered Set of Topology
Let $\left({S, \tau}\right)$ be a topological space. Let $\left({\tau, \preceq}\right)$ be an ordered set where $\preceq \mathop = \subseteq\restriction_{\tau \times \tau}$ Let $x, y \in \tau$. Then :$x \ll y$ in $\left({\tau, \preceq}\right)$  :for every set $F$ of open subsets of $S$: if $y \subseteq \bigcup F$, then ::there exists a finite subset $G$ of $F$: $x \subseteq \bigcup G$ where $\ll$ denotes the way below relation.
Continuous Inverse Theorem
Let $f$ be a real function defined on an interval $I$. Let $f$ be strictly monotone and continuous on $I$. Let $g$ be the inverse mapping to $f$. Let $J := f \left[{I}\right]$ be the image of $I$ under $f$. Then $g$ is strictly monotone and continuous on $J$.
Logarithm is Strictly Concave
:$\ln x: x > 0$ strictly concave.
Exponential is Strictly Convex
:The function $f \left({x}\right) = \exp x$ is strictly convex.
Way Below in Complete Lattice
Let $\left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $x, y \in S$. Then :$x \ll y$  :$\forall X \subseteq S: y \preceq \sup X \implies \exists A \in {\it Fin}\left({X}\right): x \preceq \sup A$ where :$\ll$ denotes the way below relation, :${\it Fin}\left({X}\right)$ denotes the set of all finite subsets of $X$.
Continuous Midpoint-Convex Function is Convex
Let $f$ be a real function which is defined on a real interval $I$. Let $f$ be midpoint-convex and continuous on $I$. Then $f$ is convex.
Power Function on Strictly Positive Base is Continuous
Let $a \in \R_{>0}$.
Continuous Strictly Midpoint-Concave Function is Strictly Concave
Let $f$ be strictly midpoint-concave and continuous on $I$. Then $f$ is strictly concave.
Continuous Midpoint-Concave Function is Concave
Let $f$ be midpoint-concave and continuous on $I$. Then $f$ is concave.
Continuous Strictly Midpoint-Convex Function is Strictly Convex
Let $f$ be strictly midpoint-convex and continuous on $I$. Then $f$ is strictly convex.
Power Function on Strictly Positive Base is Convex
Let $a \in \R_{>0}$ be a strictly positive real number. Let $f: \R \to \R$ be the real function defined as: :$\map f x = a^x$ where $a^x$ denotes $a$ to the power of $x$. Then $f$ is convex.
Natural Logarithm as Derivative of Exponential at Zero
Let $\ln: \R_{>0}$ denote the real natural logarithm. Then: :$\displaystyle \forall x \in \R_{>0}: \ln x = \lim_{h \mathop \to 0} \frac {x^h - 1} h$
Product of Increasing Positive Functions is Increasing
Let $f$ and $g$ be real functions defined on an interval $I$. Let $f$ and $g$ be increasing and positive on $I$. Then the product $f g$ is increasing on $I$.
Derivative of Exponential Function/Proof 5/Lemma
:$\forall x \in \R : n \ge \left\lceil{\left\vert{x}\right\vert}\right\rceil \implies \left\langle{\dfrac n {n + x} \left({1 + \dfrac x n}\right)^n}\right\rangle$ is increasing.
Change of Index Variable of Product
:$\displaystyle \prod_{R \left({i}\right)} a_i = \prod_{R \left({j}\right)} a_j$
Translation of Index Variable of Summation/Corollary
:$\displaystyle \sum_{j \mathop = m}^n a_j = \sum_{j \mathop = m + c}^{n + c} a_{j - c}$ where $c$ is an integer constant which is not dependent upon $j$.
Permutation of Indices of Product
:$\displaystyle \prod_{\map R j} a_j = \prod_{\map R {\map \pi j} } a_{\map \pi j}$
Translation of Index Variable of Product
:$\displaystyle \prod_{R \left({j}\right)} a_j = \prod_{R \left({c \mathop + j}\right)} a_{c \mathop + j} = \prod_{R \left({c \mathop - j}\right)} a_{c \mathop - j}$ where: :$\displaystyle \prod_{R \left({j}\right)} a_j$ denotes the product over $a_j$ for all $j$ that satisfy the propositional function $R \left({j}\right)$ :$c$ is an integer constant which is not dependent upon $j$.
Exchange of Order of Product
:$\displaystyle \prod_{R \left({i}\right)} \prod_{S \left({j}\right)} a_{i j} = \prod_{S \left({j}\right)} \prod_{R \left({i}\right)} a_{i j}$