hcju/proofwikips huggingface dataset:
Equivalence of Definitions of Connected Topological Space/No Continuous Surjection to Discrete Two-Point Space implies No Separation
Let $T = \left({S, \tau}\right)$ be a topological space. Let $T$ be such that there exists no continuous surjection from $T$ onto a discrete two-point space. Then there exist no open sets $A, B \in \tau$ such that $A, B \ne \varnothing$, $A \cup B = S$ and $A \cap B = \varnothing$.
Existence of Vector Space Bases implies Axiom of Choice
The supposition that every vector space has a basis, along with the Zermelo-Fraenkel axioms, implies that the axiom of choice holds.
Category of Posets is Cartesian Closed
Let $\mathbf{OrdSet}$ be the category of ordered sets. Then $\mathbf{OrdSet}$ is Cartesian closed.
Definition:Exponentiation Functor
Let $\mathbf C$ be a Cartesian closed metacategory. Let $A$ be an object of $\mathbf C$. Then '''exponentiation by $A$''', denoted $\left({-}\right)^A: \mathbf C \to \mathbf C$, is the functor defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = C^A := C^A        |rc= $C^A$ is the exponential of $C$ by $A$ }} {{axiom|lc= Morphism functor:        |m = f^A := \widetilde{\left({f \circ \epsilon}\right)}        |rc= $f: B \to C$ is a morphism of $\mathbf C$ }} {{end-axiom}} Here $\epsilon: B^A \times A \to B$ denotes the evaluation morphism, and $\widetilde{\left({f \circ \epsilon}\right)}: B^A \to C^A$ is the exponential transpose of $f \circ \epsilon$. That it is in fact a functor is shown on Exponentiation Functor is Functor.
Exponentiation Functor is Functor
Let $\mathbf C$ be a Cartesian closed metacategory. Let $A$ be an object of $\mathbf C$. Let $\left({-}\right)^A: \mathbf C \to \mathbf C$ be the exponentiation functor. Then $\left({-}\right)^A$ is a functor.
Cosine Exponential Formulation
:$\cos z = \dfrac {\map \exp {i z} + \map \exp {-i z} } 2$
Reciprocal of i
Let $i$ be the imaginary unit such that $i^2 = -1$. Then $\dfrac 1 i = -i$.
Squeeze Theorem/Functions/Proof 3
Let $a$ be a point on an open real interval $I$. Also let $f$, $g$ and $h$ be real functions defined at all points of $I$ except for possibly at point $a$. Suppose that: : $\forall x \ne a \in I: g \left({x}\right) \le f \left({x}\right) \le h \left({x}\right)$ : $\displaystyle \lim_{x \mathop \to a} \ g \left({x}\right) = \lim_{x \mathop \to a} \ h \left({x}\right) = L$ Then: : $\displaystyle \lim_{x \mathop \to a} \ f \left({x}\right) = L$
Rational Numbers are not Connected
The set of rational numbers $\Q$ is not a connected topological space.
Closure of Connected Set is Connected
Let $T$ be a topological space. Let $H$ be a connected set of $T$. Let $H^-$ denote the closure of $H$ in $T$.  Then $H^-$ is connected in $T$.
Closed Topologist's Sine Curve is Connected
Let $G$ be the graph of the function $y = \map \sin {\dfrac 1 x}$ for $x > 0$. Let $J$ be the line segment joining the points $\tuple {0, -1}$ and $\tuple {0, 1}$ in $\R^2$. Then $G \cup J$ is connected.
Separated Sets are Clopen in Union
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ and $B$ be separated sets in $T$. Let $H = A \cup B$ be given the subspace topology. Then $A$ and $B$ are each both open and closed in $H$.
Compatibility of Atlases is Equivalence Relation
Let $M$ be a topological space. Let $d$ and $k$ be natural numbers. Let $\mathcal A$ denote the set of all $d$-dimensional atlases of class $\mathcal C^k$ on $M$. Define a relation $\sim$ on $\mathcal A$ by putting, for any two $\mathcal C^k$-atlases $\mathcal F$ and $\mathcal G$: :$\mathcal F \sim \mathcal G$  $\mathcal F$ and $\mathcal G$ are $C^k$-compatible. Then $\sim$ is an equivalence relation on $\mathcal A$.
Addition of Real and Imaginary Parts
Let $z_0, z_1 \in \C$ be two complex numbers. Then: :$\map \Re {z_0 + z_1} = \map \Re {z_0} + \map \Re {z_1}$ and: :$\map \Im {z_0 + z_1} = \map \Im {z_0} + \map \Im {z_1}$ Here, $\map \Re {z_0}$ denotes the real part of $z_0$, and $\map \Im {z_0}$ denotes the imaginary part of $z_0$.
Multiplication of Real and Imaginary Parts
Let $w, z \in \C$ be complex numbers. $(1)$ If $w$ is wholly real, then: :$\map \Re {w z} = w \, \map \Re z$ and: :$\map \Im {w z} = w \, \map \Im z$ $(2)$ If $w$ is wholly imaginary, then: :$\map \Re {w z} = -\map \Im w \, \map \Im z$ and: :$\map \Im {w z} = \map \Im w \, \map \Re z$ Here, $\map \Re z$ denotes the real part of $z$, and $\map \Im z$ denotes the imaginary part of $z$.
Rational Numbers are Totally Disconnected
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is a totally disconnected space.
Rational Numbers are not Discrete Space
The set of rational numbers $\Q$ does not form a discrete space.
Rational Numbers are not Extremally Disconnected
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ does not form an extremally disconnected space.
Real and Imaginary Part Projections are Continuous
Define the real-valued functions $x, y: \C \to \R$ by: :$\forall z \in \C: \map x z = \map \Re z$ :$\forall z \in \C: \map y z = \map \Im z$ Equip $\R$ with the usual Euclidean metric. Equip $\C$ with the usual Euclidean metric. {{refactor|Reconsider the above link so that it goes to an actual definition page.}} Then both $x$ and $y$ are continuous functions.
Continuous Complex Function is Complex Riemann Integrable
Let $\closedint a b$ be a closed real interval. Let $f: \closedint a b \to \C$ be a continuous complex function. Then $f$ is complex Riemann integrable over $\closedint a b$.
Compact Subspace of Metric Space is Sequentially Compact in Itself
Let $M = \struct {A, d}$ be a metric space. Let $C \subseteq M$ be a subspace of $M$ such that $C$ is compact. Then $C$ is sequentially compact in itself.
Sum of Complex Integrals on Adjacent Intervals
Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval. Let $f: \left[{a \,.\,.\, b}\right] \to \C$ be a continuous complex function. Let $c \in \left[{a \,.\,.\, b}\right]$. Then: :$\displaystyle \int_a^c f \left({t}\right) \ \mathrm dt + \int_c^b f \left({t}\right) \ \mathrm dt = \int_a^b f \left({t}\right) \ \mathrm dt$
Equivalence of Definitions of Metric Space Continuity at Point
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$. Let $a \in A_1$ be a point in $A_1$. {{TFAE|def = Continuous at Point of Metric Space|continuity at a point}}
Ordering Principle
Let $S$ be a set. Then there exists a total ordering on $S$.
Number Smaller than Lebesgue Number is also Lebesgue Number
Let $M = \left({A, d}\right)$ be a metric space. Let $\epsilon \in \R_{>0}$ be a Lebesgue number for $M$. Let $\epsilon' \in \R_{>0}: \epsilon' < \epsilon$. Then $\epsilon'$ is also a Lebesgue number for $M$.
Open Cover may not have Lebesgue Number
Let $M = \left({A, d}\right)$ be a metric space. Let $\mathcal C$ be an open cover of $M$. Then it may not necessarily be the case that $\mathcal C$ has a Lebesgue number.
Burnside's Theorem
Let $G$ be a finite group. Let the order of $G$ be $p^m q^n$ where: : $p, q$ are prime : $m, n \in \N$ Then $G$ is solvable.
Constant Function is Uniformly Continuous/Metric Space
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $f_c: A_1 \to A_2$ be the constant mapping from $A_1$ to $A_2$: :$\exists c \in A_2: \forall a \in A_1: f_c \left({a}\right) = c$ That is, every point in $A_1$ maps to the same point $c$ in $A_2$. Then $f_c$ is uniformly continuous throughout $A_1$ with respect to $d_1$ and $d_2$.
Constant Function is Uniformly Continuous/Real Function
Let $f_c: \R \to \R$ be the constant mapping: :$\exists c \in \R: \forall a \in \R: \map {f_c} a = c$ Then $f_c$ is uniformly continuous on $\R$.
Linear Combination of Complex Integrals
Let $\closedint a b$ be a closed real interval. Let $f, g: \closedint a b \to \C$ be complex Riemann integrable functions over $\closedint a b$. Let $\lambda, \mu \in \C$ be complex constants. Then: :$\displaystyle \int_a^b \paren {\lambda \map f t + \mu \map g t} \rd t = \lambda \int_a^b \map f t \rd t + \mu \int_a^b \map g t \rd t$
Modulus Larger than Real Part and Imaginary Part
Let $z \in \C$ be a complex number. Let $\operatorname{Re} \left({z}\right)$ denote the real part of $z$, and $\operatorname{Im} \left({z}\right) $ the imaginary part of $z$. Then:
Pointwise Difference of Simple Functions is Simple Function
Let $\left({X, \Sigma}\right)$ be a measurable space. Let $f,g : X \to \R$ be simple functions. Then $f - g: X \to \R, \left({f - g}\right) \left({x}\right) := f \left({x}\right) - g \left({x}\right)$ is also a simple function.
Modulus and Argument of Complex Exponential
Let $z \in \C$ be a complex number. Let $\hointr a {a + 2 \pi}$ be a half open interval of length $2 \pi$. Let $r \in \hointr 0 {+\infty}$ and $\theta \in \hointr a {a + 2 \pi}$. Then: :$r = \cmod z$ and $\theta = \map \arg z$ : :$z = r e^{i \theta}$ where: :$\cmod z$ denotes the modulus of $z$ :$\map \arg z$ denotes the argument of $z$ :$x \mapsto e^x$ is the complex exponential function. If $z = 0$ or $r = 0$, then $\theta$ may be any number in $\hointr a {a + 2 \pi}$.
Bases of Vector Space have Equal Cardinality
Let $k$ be a division ring. Let $V$ be a vector space over $k$. Let $X$ and $Y$ be bases of $V$. Then $X$ and $Y$ are equinumerous.
Equivalence of Definitions of Symmetric Difference/(1) iff (2)
Let $S$ and $T$ be sets. {{TFAENocat|def = Symmetric Difference|view = symmetric difference $S * T$ between $S$ and $T$}}
Equivalence of Definitions of Symmetric Difference/(1) iff (3)
Let $S$ and $T$ be sets. {{TFAENocat|def = Symmetric Difference|view = symmetric difference $S * T$ between $S$ and $T$}}
Equivalence of Definitions of Symmetric Difference/(2) iff (4)
Let $S$ and $T$ be sets. {{TFAENocat|def = Symmetric Difference|view = symmetric difference $S * T$ between $S$ and $T$}}
Subset of Countable Set is Countable
A subset of a countable set is countable.
Sine Exponential Formulation
:$\sin z = \dfrac {\map \exp {i z} - \map \exp {-i z} } {2 i}$
Sine and Cosine Exponential Formulation
For any complex number $x$:
Sine of Sum
:$\map \sin {a + b} = \sin a \cos b + \cos a \sin b$
Hyperbolic Sine Function is Odd
:$\map \sinh {-x} = -\sinh x$
Exponential of Sum/Complex Numbers
Let $z_1, z_2 \in \C$ be complex numbers. Let $\exp z$ be the exponential of $z$. Then: :$\map \exp {z_1 + z_2} = \paren {\exp z_1} \paren {\exp z_2}$
Hyperbolic Cosine Function is Even
:$\map \cosh {-x} = \cosh x$
Hyperbolic Tangent Function is Odd
Let $\tanh: \C \to \C$ be the hyperbolic tangent function on the set of complex numbers. Then $\tanh$ is odd: :$\map \tanh {-x} = -\tanh x$
Hyperbolic Sine in terms of Sine
Let $z \in \C$ be a complex number. Then: :$i \sinh z = \map \sin {i z}$ where: :$\sin$ denotes the complex sine :$\sinh$ denotes the hyperbolic sine :$i$ is the imaginary unit: $i^2 = -1$.
Hyperbolic Cosine in terms of Cosine
Let $z \in \C$ be a complex number. Then: :$\cosh z = \map \cos {i z}$ where: :$\cos$ denotes the complex cosine :$\cosh$ denotes the hyperbolic cosine :$i$ is the imaginary unit: $i^2 = -1$.
Hyperbolic Tangent in terms of Tangent
Let $z \in \C$ be a complex number. Then: :$i \tanh z = \map \tan {i z}$ where: :$\tan$ denotes the tangent function :$\tanh$ denotes the hyperbolic tangent :$i$ is the imaginary unit: $i^2 = -1$.
Order-Extension Principle
Let $S$ be a set. Let $\preceq$ be an ordering on $S$. Then there exists a total ordering $\le$ on $S$ such that: :$\forall a, b \in S: \left({a \preceq b \implies a \le b}\right)$
Union of Nest of Orderings is Ordering
Let $S$ be a set. Let $C$ be a nonempty nest of orderings on $S$. Then $\bigcup C$ is an ordering on $S$.
Ordering can be Expanded to compare Additional Pair
Let $\left({S, \preceq}\right)$ be an ordered set. Let $a$ and $b$ be non-comparable elements of $S$. That is, let: :$a \not\preceq b$ and: :$b \not\preceq a$ Let ${\preceq'} = {\preceq} \cup \left\{ {\left({a, b}\right)} \right\}$. Let $\preceq'^+$ be the transitive closure of $\preceq'$. Then: :$\preceq'^+$ is an ordering. $\preceq'^+$ can be defined by letting $p \preceq'^+ q$ : :$p \preceq q$ or :$p \preceq a$ and $b \preceq q$.
Contour Integral is Well-Defined
Let $C$ be a contour defined by a finite sequence $C_1, \ldots, C_n$ of directed smooth curves in the complex plane$\C$. Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i \,.\,.\, b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Let $f: \operatorname{Im} \left({C}\right) \to \C$ be a continuous complex function, where $\operatorname{Im} \left({C}\right)$ denotes the image of $C$. Suppose that $\sigma_i: \left[{c_i \,.\,.\, d_i}\right] \to \C$ is a reparameterization of $C_i$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Then: :$\displaystyle \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} f \left({\gamma_i \left({t}\right) }\right) \gamma_i' \left({t}\right) \rd t = \sum_{i \mathop = 1}^n \int_{c_i}^{d_i} f \left({\sigma_i \left({t}\right) }\right) \sigma_i' \left({t}\right) \rd t$ and all complex Riemann integrals in the equation are defined.
Boolean Prime Ideal Theorem
Let $\left({S, \le}\right)$ be a Boolean algebra. Let $I$ be an ideal in $S$. Let $F$ be a filter on $S$. Let $I \cap F = \varnothing$. Then there exists a prime ideal $P$ in $S$ such that: : $I \subseteq P$ and: : $P \cap F = \varnothing$
Tangent Exponential Formulation
Let $z$ be a complex number. Let $\tan z$ denote the tangent function and $i$ denote the imaginary unit: $i^2 = -1$. Then:
Arcsine Logarithmic Formulation
For any real number $x: -1 \le x \le 1$: :$\arcsin x = \dfrac 1 i \map \ln {i x + \sqrt {1 - x^2} }$ where $\arcsin x$ is the arcsine and $i^2 = -1$.
Arccosine Logarithmic Formulation
For any real number $x: -1 \le x \le 1$: :$\arccos x = \dfrac 1 i \map \ln {x + \sqrt {x^2 - 1} }$ where $\arccos x$ is the arccosine and $i^2 = -1$.
Arctangent Logarithmic Formulation
For any real number $x$: :$\arctan x = \dfrac 1 2 i \, \map \ln {\dfrac {1 - i x} {1 + i x} }$ where $\arctan x$ is the arctangent and $i^2 = -1$.
Two Ring is Boolean Ring
Let $2$ be the two ring. Then $2$ is a Boolean ring.
Idempotent Ring is Commutative
Let $\struct {R, +, \circ}$ be an idempotent ring. Denote with $0_R$ the zero of $R$. Then $\struct {R, +, \circ}$ is a commutative ring.
Idempotent Ring has Characteristic Two
Let $\struct {R, +, \circ}$ be an idempotent non-null ring. Denote with $0_R$ the zero of $R$. Then $\struct {R, +, \circ}$ has characteristic $2$.
Two-Valued Functions form Boolean Ring
Let $S$ be a set, and let $2$ be the two ring. Let $2^S$ be the set of all $2$-valued functions on $S$. Denote with $+$ and $\cdot$ the pointwise operations induced on $2^S$ by $+_2$ and $\times_2$, respectively. Then $\struct {2^S, +, \cdot}$ is a Boolean ring.
Complex Integration by Substitution
Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval. Let $\phi: \left[{a \,.\,.\, b}\right] \to \R$ be a real function which has a derivative on $\left[{a \,.\,.\, b}\right]$. Let $f: A \to \C$ be a continuous complex function, where $A$ is a subset of the image of $\phi$. If $\phi \left({a}\right) \le \phi \left({b}\right)$, then: :$\displaystyle \int_{\phi \left({a}\right)}^{\phi \left({b}\right)} f \left({t}\right) \, \mathrm d t = \int_a^b f \left({\phi \left({u}\right)}\right) \phi' \left({u}\right) \, \mathrm d u$ If $\phi \left({a}\right) > \phi \left({b}\right)$, then: :$\displaystyle \int_{\phi \left({b}\right)}^{\phi \left({a}\right)} f \left({t}\right) \, \mathrm  dt = -\int_a^b f \left({\phi \left({u}\right)}\right) \phi' \left({u}\right) ,\ \mathrm d u$
Continuous Injection of Interval is Strictly Monotone
Let $I$ be a real interval. Let $f: I \to \R$ be an injective continuous real function. Then $f$ is strictly monotone.
Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice
Let $L$ be a distributive lattice. {{explain|Full notational specification needed}} Let $F$ be a filter in $L$. Let $M$ be an ideal in $L$ which is disjoint from $F$ such that no ideal in $L$ larger than $M$ is disjoint from $F$. Then $M$ is a prime ideal.
Prime Ideal in Lattice
Let $\left({L, \le}\right)$ be a lattice. Let $I$ be an ideal in $L$. Then $I$ is a prime ideal : :$\forall a, b \in L: a \wedge b \in I \implies a \in I \text{ or } b \in I$ where $a \wedge b$ denotes $\min \left\{{a, b}\right\}$, the meet of $a$ and $b$.
Inverse of Increasing Bijection need not be Increasing
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. Let $\phi: S \to T$ be a bijection which is increasing. Then $\phi^{-1}: T \to S$ is not necessarily also increasing.
Order Isomorphic Sets are Equivalent
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be order isomorphic. Then $S$ and $T$ are equivalent.
Linear Combination of Contour Integrals
Let $C$ be a contour in $\C$. Let $f, g: \Img C \to \C$ be continuous complex functions, where $\Img C$ denotes the image of $C$. Let $\lambda, \mu \in \C$ be complex constants. Then: :$\displaystyle \int_C \paren {\lambda \map f z + \mu \map g z} \rd z = \lambda \int_C \map f z \rd z + \mu \int_C \map g z \rd z$
P-Product Metric Induces Product Topology
Let $M_A = \struct{A, d_A}$ and $M_B = \struct{B, d_B}$ be metric spaces. Let $\tau_A$ and $\tau_B$ be the topologies on $A$ and $B$ induced by $d_A$ and $d_B$, respectively. Let $p \ge 1$ be an extended real number. Let $M = \struct{A \times B, d}$ be the $p$-product of $M_A$ and $M_B$. We have that $M$ is a metric space. Let $\tau$ be the topology on $A \times B$ induced by $d$. Then $\struct{A \times B, \tau}$ is the product space of $\struct{A, \tau_A}$ and $\struct{B, \tau_B}$.
P-Product Metrics are Lipschitz Equivalent
Let $M_A = \left({A, d_A}\right)$ and $M_B = \left({B, d_B}\right)$ be metric spaces. Let $\tau_A$ and $\tau_B$ be the topologies on $A$ and $B$ induced by $d_A$ and $d_B$, respectively. For all extended real numbers $p \ge 1$, let $M_p = \left({A \times B, d_p}\right)$ be the $p$-product of $M_A$ and $M_B$. Then all the metrics $\left\{{d_p: p \ge 1}\right\}$ are Lipschitz equivalent.
Ordered Sum of Tosets is Totally Ordered Set/General Result
Let $S_1, S_2, \ldots, S_n$ all be tosets. Let $T_n$ be the ordered sum of $S_1, S_2, \ldots, S_n$: :$\forall n \in \N_{>0}: T_n = \begin{cases} S_1 & : n = 1 \\ T_{n-1} + S_n & : n > 1 \end{cases}$ Then $T_n$ is a toset.
Supremum of Suprema
Let $\struct {S, \preceq}$ be an ordered set. Let $\mathbb T \subseteq \powerset S$, where $\powerset S$ is the power set of $S$. Suppose all $T \in \mathbb T$ admit a supremum $\sup T$ in $S$. Then: :$\sup \bigcup \mathbb T = \sup {\set {\sup T: T \in \mathbb T} }$ if one of these two quantities exists (in $S$).
Ordered Product of Tosets is Totally Ordered Set/General Result
Let $S_1, S_2, \ldots, S_n$ all be tosets. Let $T_n$ be the ordered product of $S_1, S_2, \ldots, S_n$: :$\forall n \in \N_{>0}: T_n = \begin{cases} S_1 & : n = 1 \\ T_{n-1} \cdot S_n & : n > 1 \end{cases}$ Then $T_n$ is a toset.
Derivative of Complex Composite Function
Let $f: D \to \C$ be a complex-differentiable function, where $D \subseteq \C$ is an open set. Let $g: \operatorname{Im} \left({f}\right) \to \C$ be a complex-differentiable function, where $\operatorname{Im} \left({f}\right)$ denotes the image of $f$. Define $h = f \circ g: D \to C$ as the composite of $f$ and $g$. Then $h$ is complex-differentiable, and its derivative is defined as: :$\forall z \in D: h' \left({z}\right) =  f' \left({g \left({z}\right)}\right) g' \left({z}\right)$
Ring of Idempotents is Idempotent Ring
Let $\struct {R, +, \circ}$ be a commutative ring. Let $\struct {A, \oplus, \circ}$ be its ring of idempotents. Then $\struct {A, \oplus, \circ}$ is an idempotent ring.
Kelley's Theorem
Let $\struct {D, \preceq}$ be a directed set, Let $S$ be a non-empty set. Let $n: D \to S$ be a net in $S$. Then $n$ has a universal subnet.
Contour Integral is Independent of Parameterization
Let $C$ be a contour defined by a finite sequence $C_1, \ldots, C_n$ of directed smooth curves. Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i \,.\,.\, b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Let $f: \operatorname{Im} \left({C}\right) \to \C$ be a continuous complex function, where $\operatorname{Im} \left({C}\right)$ denotes the image of $C$. Suppose that $\sigma_i: \left[{c_i \,.\,.\, d_i}\right] \to \C$ is a reparameterization of $C_i$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Then: :$\displaystyle \int_C f \left({z}\right) \rd z = \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} f \left({\gamma_i \left({t}\right) }\right) \gamma_i' \left({t}\right) \rd t = \sum_{i \mathop = 1}^n \int_{c_i}^{d_i} f \left({\sigma_i \left({t}\right) }\right) \sigma_i' \left({t}\right) \rd t$
Unity is Unity in Ring of Idempotents
Let $\left({R, +, \circ}\right)$ be a commutative and unitary ring whose unity is $1_R$. Let $\left({A, \oplus, \circ}\right)$ be the ring of idempotents of $R$. Then $1_R$ is also a unity for $\left({A, \oplus, \circ}\right)$.
Ring of Idempotents of Commutative and Unitary Ring is Boolean Ring
Let $\struct {R, +, \circ}$ be a commutative and unitary ring. Let $\struct {A, \oplus, \circ}$ be its ring of idempotents. Then $\struct {A, \oplus, \circ}$ is a Boolean ring.
Definition:Boolean Group
Let $\struct {G, \circ}$ be a group. Then $\struct {G, \circ}$ is a '''Boolean group'''  all its elements, other than the identity, have order $2$.
Boolean Ring has Proper Zero Divisor
Let $\left({R, +, \circ}\right)$ be a Boolean ring whose zero is $0_R$. Suppose that $R$ has more than two elements. Then $R$ has a proper zero divisor.
Continuous Real Function on Closed Interval is Bijective iff Strictly Monotone
Let $\closedint a b$ and $\closedint c d$ be closed real intervals. Let $f: \closedint c d \to \closedint a b$ be a continuous real function. Let $\map f c, \map f d \in \set {a, b}$. Then $f$ is bijective  $f$ is strictly monotone.
Concatenation of Contours is Contour
Let $C$ and $D$ be contours. That is, $C$ is a finite sequence of directed smooth curves $C_1, \ldots, C_n$. Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i\,.\,.\,b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Similarly, $D$ is a finite sequence of directed smooth curves $D_1, \ldots, D_m$. Let $D_i$ be parameterized by the smooth path $\sigma_i: \left[{c_i\,.\,.\,d_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, m}\right\}$. Suppose $\gamma_n \left({b_n}\right) = \sigma_1 \left({c_1}\right)$. Then the finite sequence: :$C_1, \ldots, C_n, D_1, \ldots, D_m$ defines a contour.
Contour Integral of Concatenation of Contours
Let $C$ and $D$ be contours in $\C$. Suppose that the end point of $C$ is equal to the start point of $D$, so the concatenation $C \cup D$ is defined. Let $f: \operatorname{Im} \left({C \cup D}\right) \to \C$ be a continuous complex function, where $\operatorname{Im} \left({C \cup D}\right)$ denotes the image of $C \cup D$. Then: :$\displaystyle \int_{C \cup D} f \left({z}\right) \ \mathrm dz = \int_C f \left({z}\right) \rd z + \int_D f \left({z}\right) \rd z$
Regular Expression is Accepted by Finite State Machine
Let $R$ be a regular expression. Then there exists a finite state machine $F$ such that its accepted language $L \left({F}\right)$ is exactly $L \left({R}\right)$, the language defined by $R$.
Directed Smooth Curve Relation is Equivalence
Let $\sim$ denote a relation on the set of all smooth paths: $\left\{ {\gamma: I \to \C : \text{$I$ is a closed real interval, $\gamma$ is a smooth path} }\right\}$. Let $\gamma : \left[{ a \,.\,.\, b }\right] \to \C$ and $\sigma : \left[{c \,.\,.\, d}\right] \to \C$ be smooth paths. Define $\sim$ as follows:  :$\gamma \sim \sigma$  there exists a bijective differentiable strictly increasing real function $\phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$ such that $\sigma = \gamma \circ \phi$. Then $\sim$ is an equivalence relation on the set of all smooth paths.