avewright/proofwiki-math huggingface dataset:
Theorem: Topological Properties of Non-Archimedean Division Rings/Intersection of Closed Balls

:$\map { {B_r}^-} x \cap \map { {B_s}^-} y \ne \O \iff \map { {B_r}^-} x \subseteq \map { {B_s}^-} y$ or $\map { {B_s}^-} y \subseteq \map { {B_r}^-} x$

Proof:
=== Necessary Condition ===
Let $z \in \map { {B_r}^-} x \cap \map { {B_s}^-} y$.
If $r \le s$ then:
{{begin-eqn}}
{{eqn| l = \map { {B_r}^-} x 
     | r = \map { {B_r}^-} z
     | c = Every element in an open ball is the center
}}
{{eqn| o = \subseteq
     | r = \map { {B_s}^-} z
     | c = as $r \le s$
}}
{{eqn| r = \map { {B_s}^-} y
     | c = Every element in an open ball is the center
}}
{{end-eqn}}
Similarly, if $s \le r$ then:
{{begin-eqn}}
{{eqn| l = \map { {B_s}^-} y 
     | o = \subseteq
     | r = \map { {B_r}^-} x
}}
{{end-eqn}}
{{qed|lemma}}
Theorem: Mittag-Leffler Expansion for Hyperbolic Cotangent Function

:$\ds \pi \map \coth {\pi z} = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 + n^2}$
where: 
:$z \in \C$ is not an integer multiple of $i$
:$\coth$ is the hyperbolic cotangent function.

Proof:
{{begin-eqn}}
{{eqn	| l = \pi \map \coth {\pi z}
	| r = \pi i \map \cot {\pi i z}
	| c = Hyperbolic Cotangent in terms of Cotangent
}}
{{eqn	| r = i \paren {\frac 1 {i z} + 2 i \sum_{n \mathop = 1}^\infty \frac z {\paren {i z}^2 - n^2} }
	| c = Mittag-Leffler Expansion for Cotangent Function
}}
{{eqn	| r = \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \frac z {-z^2 - n^2}
	| c = $i^2 = -1$
}}
{{eqn	| r = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 + n^2}
}}
{{end-eqn}}
{{qed}}
{{Namedfor|Magnus Gustaf Mittag-Leffler|cat = Mittag-Leffler}}
Theorem: Mittag-Leffler Expansion for Secant Function

:$\ds \pi \map \sec {\pi z} = 4 \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {2 n + 1} {\paren {2 n + 1}^2 - 4 z^2}$
where: 
:$z \in \C$ is not a half-integer
:$\sec$ is the secant function.

Proof:
{{begin-eqn}}
{{eqn | l = \pi \map \sec {\pi z}
      | r = \pi \map \csc {\frac \pi 2 - \pi z}
      | c = Secant and Cosecant are Cofunctions in radians
}}
{{eqn | r = \frac 1 {1/2 - z} + 2 \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {1/2 - z} {\paren {1/2 - z}^2 - n^2}
      | c = Mittag-Leffler Expansion for Cosecant Function
}}
{{eqn | r = \frac 1 {1/2 - z} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \paren {\frac 1 {n + 1/2 - z} - \frac 1 {n - 1/2 + z} }
      | c = Partial Fractions Expansion
}}
{{eqn | r = \frac 1 {1/2 - z} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac 1 {n + 1/2 - z} + \sum_{n \mathop = 1}^\infty \paren {-1}^{n + 1} \frac 1 {n - 1/2 + z}
      | c = splitting into two convergent series
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {n + 1/2 - z} + \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {n + 1/2 + z}
      | c = rewriting both sums starting from $n=0$
}}
{{eqn | r = 2 \sum_{n \mathop = 0}^\infty \paren {-1}^n \paren {\frac 1 {2 n - 2 z + 1} + \frac 1 {2 n + 2 z + 1} }
      | c = combining convergent series
}}
{{eqn | r = 4 \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {2 n + 1} {\paren {2 n + 1}^2 - 4 z^2}
      | c = Difference of Two Squares
}}
{{end-eqn}}
{{qed}}
{{Namedfor|Magnus Gustaf Mittag-Leffler|cat = Mittag-Leffler}}
Theorem: Pringsheim's Theorem

Let $f$ be a holomorphic function defined on a unit disc centered at the origin of the complex plane and is denoted by its Taylor series:
:$\map f z = \ds \sum_{n \mathop = 0}^{\infty} c_n z^n$
Let:
:$(1): \quad \forall n \ge 0: c_n \ge 0$
:$(2): \quad$ the radius of convergence of the Taylor series of function $f$ is $1$.
Then $z = 1$ is an isolated singularity of $f$.

Proof:
{{ProofWanted}}
{{Namedfor|Alfred Pringsheim|cat = Pringsheim}}
ljnkhn1tkbziuxpfgxvxwj0cle7jp03
Theorem: Mittag-Leffler Expansion for Tangent Function

:$\ds \pi \map \tan {\pi z} = 8 \sum_{n \mathop = 0}^\infty \frac z {\paren {2 n + 1}^2 - 4 z^2}$
where: 
:$z \in \C$ is not a half-integer
:$\tan$ is the tangent function.

Proof:
From {{Corollary|Mittag-Leffler Expansion for Tangent Function}}, we have:
{{begin-eqn}}
{{eqn | l = \frac \pi {2 n} \map \tan {\frac {\pi m} {2 n} }
      | r = \sum_{k \mathop = 0}^\infty \paren {\frac 1 {\paren {2 k + 1} n - m} - \frac 1 {\paren {2 k + 1} n + m} }
      | c = 
}}
{{eqn | r = \paren {\frac 1 {n - m} - \frac 1 {n + m} } + \paren {\frac 1 {3 n - m} - \frac 1 {3 n + m} } + \paren {\frac 1 {5 n - m} - \frac 1 {5 n + m} } + \cdots
      | c = 
}}
{{end-eqn}}
Let $m := 1$ and $n := 2$.
Then:
{{begin-eqn}}
{{eqn | l = \frac \pi 4 \map \tan {\frac \pi 4}
      | r = \sum_{k \mathop = 0}^\infty \paren {\frac 1 {4 k + 1} - \frac 1 {4 k + 3} }
      | c = 
}}
{{eqn | ll = \leadsto
      | l = \frac \pi 4
      | r = \paren {\frac 1 1 - \frac 1 3} + \paren {\frac 1 5 - \frac 1 7} + \paren {\frac 1 9 - \frac 1 {11} } + \cdots
      | c = {{tan|45}}
}}
{{end-eqn}}
{{qed}}
Theorem: Mittag-Leffler Expansion for Tangent Function

:$\ds \pi \map \tan {\pi z} = 8 \sum_{n \mathop = 0}^\infty \frac z {\paren {2 n + 1}^2 - 4 z^2}$
where: 
:$z \in \C$ is not a half-integer
:$\tan$ is the tangent function.

Proof:
{{ProofWanted}}
{{Namedfor|Magnus Gustaf Mittag-Leffler|cat = Mittag-Leffler}}
Theorem: Mittag-Leffler Expansion for Hyperbolic Tangent Function

:$\ds \pi \map \tanh {\pi z} = 8 \sum_{n \mathop = 0}^\infty \frac z {4 z^2 + \paren {2 n + 1}^2}$
where:
:$z \in \C$ is not a half-integer multiple of $i$
:$\tanh$ is the hyperbolic tangent function.

Proof:
{{ProofWanted}}
{{Namedfor|Magnus Gustaf Mittag-Leffler|cat = Mittag-Leffler}}
Theorem: Mittag-Leffler Expansion for Hyperbolic Secant Function

:$\ds \pi \map \sech {\pi z} = 4 \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {2 n + 1} {\paren {2 n + 1}^2 + 4 z^2}$
where: 
:$z \in \C$ is not a half-integer multiple of $i$ 
:$\sech$ is the hyperbolic secant function.

Proof:
{{begin-eqn}}
{{eqn | l = \pi \map \sech {\pi z}
      | r = \pi \map \sec {i \pi z}
      | c = Hyperbolic Secant in terms of Secant
}}
{{eqn | r = 4 \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {2 n + 1} {\paren {2 n + 1}^2 - 4 \paren {i z}^2}
      | c = Mittag-Leffler Expansion for Secant Function
}}
{{eqn | r = 4 \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {2 n + 1} {\paren {2 n + 1}^2 + 4 z^2}
      | c = $i^2 = -1$
}}
{{end-eqn}}
{{qed}}
{{Namedfor|Magnus Gustaf Mittag-Leffler|cat = Mittag-Leffler}}
Theorem: Mittag-Leffler Expansion for Hyperbolic Cosecant Function

:$\ds \pi \map \csch {\pi z} = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac z {z^2 + n^2}$
where:
:$z \in \C$ is not an integer multiple of $i$
:$\csch$ is the hyperbolic cosecant function.

Proof:
{{begin-eqn}}
{{eqn | l = \pi \map \csch {\pi z}
      | r = i \pi \map \csc {i \pi z}
      | c = Hyperbolic Cosecant in terms of Cosecant
}}
{{eqn | r = i \paren {\dfrac 1 {i z} + 2 i \sum _{n \mathop = 1}^\infty \paren {-1}^n \dfrac z {\paren {i z}^2 - n^2} }
      | c = Mittag-Leffler Expansion for Cosecant Function
}}
{{eqn | r = \dfrac 1 z - 2 \sum_{n \mathop = 1}^\infty \paren {-1}^n \dfrac z {-z^2 - n^2}
      | c = $i^2=-1$
}}
{{eqn | r = \dfrac 1 z + 2 \sum_{n \mathop = 1}^\infty \paren {-1}^n \dfrac z {z^2 + n^2}
}}
{{end-eqn}}
{{qed}}
{{Namedfor|Magnus Gustaf Mittag-Leffler|cat = Mittag-Leffler}}
Theorem: Subtraction has no Identity Element

The operation of subtraction on numbers of any kind has no identity.

Proof:
{{AimForCont}} there exists an identity $e$ in one of the standard number systems $\GF$.
{{begin-eqn}}
{{eqn | q = \forall x \in \GF
      | l = x
      | r = x - e
      | c = 
}}
{{eqn | r = e - x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = x + \paren {-e}
      | r = e + \paren {-x}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = x + x
      | r = e + e
      | c = 
}}
{{eqn | ll= \leadsto
      | l = x
      | r = e
      | c = 
}}
{{end-eqn}}
That is:
:$\forall x \in \GF: x = e$
But from Identity is Unique, if $e$ is an identity then there can be only one such.
From Proof by Contradiction it follows that $\GF$ has no such $e$.
{{qed}}
Theorem: Zero Element of Multiplication on Numbers

On all the number systems:
* natural numbers $\N$
* integers $\Z$
* rational numbers $\Q$
* real numbers $\R$
* complex numbers $\C$
the zero element of multiplication is zero ($0$).

Proof:
This is demonstrated by showing that:
:$n \times 0 = 0 = 0 \times n$
for all $n$ in all standard number systems.
{{qed}}
Theorem: Addition on Numbers has no Zero Element

On all the number systems:
* natural numbers $\N$
* integers $\Z$
* rational numbers $\Q$
* real numbers $\R$
* complex numbers $\C$
there exists no zero element for addition.

Proof:
Suppose $z$ is a zero element for addition in a standard number system $\F$.
Then:
{{begin-eqn}}
{{eqn | q = \forall n \in \F
      | l = n + z
      | r = z
      | c = 
}}
{{eqn | ll= \leadsto
      | l = n
      | r = 0
      | c = subtracting $z$ from both sides
}}
{{end-eqn}}
As $n$ is arbitrary, and therefore not always $0$, it follows there can be no such $z$.
{{qed}}
Theorem: Bernstein's Theorem on Unique Global Solution to y''=F(x,y,y')

Let $F$ and its partial derivatives $F_y, F_{y'}$ be real functions, defined on the closed interval $I = \closedint a b$.
Let $F, F_y, F_{y'} $ be continuous at every point $\tuple {x, y}$ for all finite $y'$.
Suppose there exists a constant $k > 0$ such that:
:$\map {F_y} {x, y, y'} > k$
Suppose there exist real functions $\alpha = \map \alpha {x, y} \ge 0$, $\beta = \map \beta {x, y}\ge 0$ bounded in every bounded region of the plane such that:
:$\size {\map F {x, y, y'} } \le \alpha y'^2 + \beta$
Then one and only one integral curve of the equation $y'' = \map F {x, y, y'}$ passes through any two points $\tuple {a, A}$ and $\tuple {b, B}$ such that $a \ne b$.

Proof:
=== Lemma 1 (Uniqueness) ===
{{:Bernstein's Theorem on Unique Global Solution to y''=F(x,y,y')/Lemma 1}}{{qed|lemma}}
Theorem: Real Numbers under Subtraction do not form Semigroup

The set of real numbers under subtraction $\struct {\R, -}$ does not form a semigroup.

Proof:
We have that Subtraction on Numbers is Not Associative.
Hence $\struct {\R, -}$ is not a semigroup by definition.
{{qed}}
Theorem: Group has Latin Square Property/Additive Notation

Let $\struct {G, +}$ be a group.
Then $G$ satisfies the Latin square property.
That is, for all $a, b \in G$, there exists a unique $g \in G$ such that $a + g = b$.
Similarly, there exists a unique $h \in G$ such that $h + a = b$.

Proof:
From Group has Latin Square Property, we have that:
{{begin-eqn}}
{{eqn | l = a + g
      | r = b
      | c = 
}}
{{eqn | ll= \leadsto
      | l = g
      | r = \paren {-a} + b
      | c = 
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn | l = h + a
      | r = b
      | c = 
}}
{{eqn | ll= \leadsto
      | l = h
      | r = b + \paren {-a}
      | c = 
}}
{{end-eqn}}
{{qed}}
Theorem: Normed Division Ring Operations are Continuous/Addition

:$+ : \struct {R \times R, d_p} \to \struct{R,d}$ 
is continuous.

Proof:
By $p$-Product Metric Induces Product Topology and Continuous Mapping is Continuous on Induced Topological Spaces, it suffices to consider the case $p = \infty$.
Let $\tuple {x_0, y_0} \in R \times R$.
Let $\epsilon > 0$ be given.
Let $\tuple {x, y} \in R \times R$ such that:
:$\map {d_\infty} {\tuple {x, y}, \tuple{x_0, y_0} } < \dfrac \epsilon 2$
By the definition of the product metric $d_\infty$ then:
:$\max \set {\map d {x, x_0}, \map d {y, y_0} } < \dfrac \epsilon 2$
or equivalently:
:$\map d {x, x_0} < \dfrac \epsilon 2$
:$\map d {y, y_0} < \dfrac \epsilon 2$
Then:
{{begin-eqn}}
{{eqn | l = \map d {x_0 + y_0, x + y}
      | r = \norm {\paren {x_0 + y_0} - \paren {x + y} }
      | c = {{Defof|Metric Induced by Norm on Division Ring}}
}}
{{eqn | r = \norm {\paren {x_0 - x} + \paren {y_0 - y} }
      | c = 
}}
{{eqn | r = \norm {x_0 - x } + \norm {y_0 - y}
      | o = \le
      | c = {{Norm-axiom-mult|3}}
}}
{{eqn | r = \map d {x_0, x} + \map d {y_0, y}
      | o = \le
      | c = {{Defof|Metric Induced by Norm on Division Ring}}
}}
{{eqn | r = \dfrac \epsilon 2 + \dfrac \epsilon 2
      | o = <
      | c = 
}}
{{eqn | r = \epsilon
}}
{{end-eqn}}
Since $\tuple {x_0, y_0}$ and $\epsilon$ were arbitrary, by the definition of continuity then the mapping:
:$+ : \struct {R \times R, d_\infty} \to \struct {R, d}$ 
is continuous.
{{qed}}
Theorem: Normed Division Ring Operations are Continuous/Negation

:$\eta: \struct {R, d} \to \struct {R, d}: \map \eta x = -x$ 
is continuous.

Proof:
Let $x_0 \in R$.
Let $\epsilon > 0$ be given.
Let $x \in R$ such that:
:$\map d {x, x_0} < \epsilon$
Then:
{{begin-eqn}}
{{eqn | l = \map d {-x, -x_0}
      | r = \norm {-x - \paren {-x_0} }
      | c = {{Defof|Metric Induced by Norm on Division Ring}}
}}
{{eqn | r = \norm {-x + x_0}
}}
{{eqn | r = \norm {x_0 - x}
      | c = $+$ is Commutative.
}}
{{eqn | r = \map d {x_0, x}
      | c = {{Defof|Metric Induced by Norm on Division Ring}}
}}
{{eqn | r = \map d {x, x_0}
      | c = {{Metric-space-axiom|2}}
}}
{{eqn | r = \epsilon
      | o = <
}}
{{end-eqn}}
Since $x_0$ and $\epsilon$ were arbitrary, by the definition of continuity then the mapping:
:$\eta: \struct {R, d} \to \struct {R, d} : \map \eta x = -x$ 
is continuous.
{{qed}}
Theorem: Normed Division Ring Operations are Continuous/Multiplication

:$* : \struct {R \times R, d_p} \to \struct {R, d}$ 
is continuous.

Proof:
By $p$-Product Metric Induces Product Topology and Continuous Mapping is Continuous on Induced Topological Spaces, it suffices to consider the case $p = \infty$.
Let $\tuple {x_0, y_0} \in R \times R$.
Let $\epsilon > 0$ be given.
Let $\delta = \min \set {\dfrac \epsilon {1 + \norm {y_0} + \norm {x_0} }, 1}$
Since $1 + \norm {y_0} + \norm {x_0} > 0$ then $\delta > 0$
Let $\tuple {x,y} \in R \times R$ such that:
:$\map {d_\infty} {\tuple {x, y}, \tuple{x_0, y_0} } < \delta$
By the definition of the $p$-product metric $d_\infty$:
:$\max \set {\map d {x, x_0}, \map d {y, y_0}} < \delta$
or equivalently:
:$\map d {x, x_0} < \delta$
:$\map d {y, y_0} < \delta$
Then:
{{begin-eqn}}
{{eqn | l = \norm y
      | r = \norm {y - y_0 + y_0}
      | c =
}}
{{eqn | o = \le
      | r = \norm {y - y_0}  + \norm {y_0}
      | c = {{Norm-axiom-mult|3}}
}}
{{eqn | o = \le
      | r = \map d {y, y_0}  + \norm {y_0}
      | c = {{Defof|Metric Induced by Norm on Division Ring}}
}}
{{eqn | o = <
      | r = \delta  + \norm {y_0}
      | c = 
}}
{{eqn | o = \le
      | r = 1  + \norm {y_0}
      | c = 
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \map d {x y, x_0 y_0}
      | r = \norm {x y - x_0 y_0}
      | c = {{Defof|Metric Induced by Norm on Division Ring}}
}}
{{eqn | r = \norm {x y - x_0 y + x_0 y - x_0 y_0}
      | c =
}}
{{eqn | o = \le
      | r = \norm {x y - x_0 y} + \norm {x_0 y - x_0 y_0}
      | c = {{Norm-axiom-mult|3}}
}}
{{eqn | o = \le
      | r = \norm {x - x_0} \norm y + \norm {x_0} \norm {y - y_0}
      | c = {{Norm-axiom-mult|2}}
}}
{{eqn | o = \le
      | r = \map d {x, x_0} \norm y + \norm {x_0} \map d {y, y_0}
      | c = {{Defof|Metric Induced by Norm on Division Ring}}
}}
{{eqn | o = <
      | r = \delta \norm y + \norm {x_0} \delta
      | c = 
}}
{{eqn | o = \le
      | r = \delta \paren {\norm y + \norm {x_0} }
      | c =
}}
{{eqn | o = <
      | r = \delta \paren {1 + \norm {y_0} + \norm {x_0} }
      | c = 
}}
{{eqn | o = \le
      | r = \dfrac \epsilon {1 + \norm {y_0} + \norm {x_0} } \paren {1 + \norm {y_0} + \norm {x_0} }
      | c = 
}}
{{eqn | r = \epsilon
}}
{{end-eqn}}
We have that $\tuple {x_0, y_0}$ and $\epsilon$ are arbitrary.
Hence, by the definition of continuity, the mapping:
:$* : \struct {R \times R, d_\infty} \to \struct {R, d}$ 
is continuous.
{{qed}}
Theorem: Normed Division Ring Operations are Continuous/Inversion

:$\iota : \struct {R^* ,d^*} \to \struct {R, d} : \map \iota x = x^{-1}$ 
is continuous.

Proof:
Let $x_0 \in R^*$.
Let $\epsilon > 0$ be given.
Let $\delta = \min \set {\dfrac {\norm {x_0} } 2, \dfrac {\norm {x_0}^2 \epsilon} 2 }$
Let $x \in R^*$ such that:
:$\map {d^*} {x, x_0} < \delta$
By the definition of the subspace metric on $R^*$ and the definition of the metric induced by the norm on $R$:
:$\map {d^*} {x, x_0} = \map d {x, x_0} = \norm {x - x_0} < \delta$
Then:
{{begin-eqn}}
{{eqn | l = \norm {x_0}
      | o = \le
      | r = \norm {x - x_0} + \norm x
      | c = {{Norm-axiom-mult|3}}
}}
{{eqn | o =
}}
{{eqn | o = <
      | r = \delta + \norm x
      | c = {{hypothesis}}: $\norm {x - x_0} < \delta$
}}
{{eqn | o =
}}
{{eqn | o = \le
      | r = \dfrac {\norm {x_0} } 2 + \norm x
      | c = {{hypothesis}}: $\delta \le \dfrac {\norm {x_0} } 2$
}}
{{eqn | o =
}}
{{eqn | ll= \leadsto
      | l = \dfrac {\norm {x_0} } 2
      | o = <
      | r = \norm x
      | c = subtracting $\dfrac {\norm {x_0} } 2$ from both sides
}}
{{eqn | o =
}}
{{eqn | ll= \leadsto
      | l = \dfrac 2 {\norm {x_0} }
      | o = >
      | r = \dfrac 1 {\norm x}
      | c = inverting both sides of the equation
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \map d {x^{-1}, x_0^{-1} }
      | r = \norm {x^{-1} - x_0^{-1} }
      | c = {{Defof|Metric Induced by Norm on Division Ring}}
}}
{{eqn | o =
}}
{{eqn | r = \dfrac 1 {\norm x} \paren {\norm x \norm {x^{-1} - x_0^{-1} } \norm {x_0} } \dfrac 1 {\norm {x_0} } 
      | c = 
}}
{{eqn | o =
}}
{{eqn | r = \dfrac 1 {\norm x \norm {x_0} } \paren {\norm {x \paren {x^{-1} - x_0^{-1} } x_0} }
      | c = {{Norm-axiom-mult|2}}
}}
{{eqn | o =
}}
{{eqn | r = \dfrac 1 {\norm x \norm {x_0} } \paren {\norm {x x^{-1} x_0 - x x_0^{-1} x_0} }
      | c = {{Ring-axiom|D}}
}}
{{eqn | o =
}}
{{eqn | r = \dfrac 1 {\norm x \norm {x_0} } \paren {\norm {x_0 - x} }
      | c = {{Defof|Division Ring}}
}}
{{eqn | o =
}}
{{eqn | o = <
      | r = \dfrac 2 {\norm {x_0}^2} \paren {\norm {x_0 - x} }
      | c = from $\dfrac 1 {\norm x} < \dfrac 2 {\norm {x_0} }$ above
}}
{{eqn | o =
}}
{{eqn | o = <
      | r = \dfrac {2 \delta} {\norm {x_0}^2}
      | c = {{hypothesis}}: $\norm {x - x_0} < \delta$
}}
{{eqn | o =
}}
{{eqn | o = \le
      | r = \dfrac 2 {\norm {x_0}^2} \paren {\dfrac {\norm {x_0}^2 \epsilon} 2}
      | c = {{hypothesis}}: $\delta \le \dfrac {\norm {x_0}^2 \epsilon} 2$
}}
{{eqn | o =
}}
{{eqn | r = \epsilon
      | c = cancelling terms
}}
{{end-eqn}}
Since $x_0$ and $\epsilon$ were arbitrary, by the definition of continuity then the mapping:
:$\iota: \struct {R^*, d^*} \to \struct {R, d} : \map \iota x = x^{-1}$ 
is continuous.
{{qed}}
Theorem: Normed Division Ring Operations are Continuous/Corollary

Let $\tau$ be the topology induced by the metric $d$.
Then:
:$\struct {R, \tau}$ is a topological division ring.

Proof:
Let $d_\infty$ be the Chebyshev distance metric on $R \times R$.
Let $\tau^\times$ be the product topology on $R \times R$.
By $p$-Product Metric Induces Product Topology, $\tau^\times$ is the topology induced by the metric $d_\infty$.
Let $R^* = R \setminus \set 0$.
Let $d^*$ be the restriction of $d$ to $R^*$.
Let $\tau^*$ be the subspace topology on $R^*$.
By Metric Subspace Induces Subspace Topology then $\tau^*$ is the topology induced by the metric $d^*$
By Normed Division Ring Operations are Continuous and Continuous Mapping is Continuous on Induced Topological Spaces, the mappings:
::$\phi : \struct {R \times R, \tau^\times} \to \struct {R, \tau} : \map \phi {x, y} = x + y$ 
::$\theta : \struct {R ,\tau} \to \struct {R, \tau} : \map \theta x = -x$ 
::$\psi : \struct {R \times R, \tau^\times} \to \struct {R, \tau} : \map \psi {x, y} = x y$ 
::$\xi : \struct {R^* ,\tau^*} \to \struct {R, \tau} : \map \xi x = x^{-1}$ 
are continuous.
By the definition of a topological division ring then the result follows.
{{qed}}
Theorem: Structure with Element both Identity and Zero has One Element

Let $\struct {S, \circ}$ be an algebraic structure.
Let $z \in S$ such that $z$ is both an identity element and a zero element.
Then:
:$S = \set z$

Proof:
Let $x \in S$.
Then
{{begin-eqn}}
{{eqn | l = x
      | r = x \circ z
      | c = {{Defof|Identity Element}}
}}
{{eqn | r = z
      | c = {{Defof|Zero Element}}
}}
{{end-eqn}}
and so there is no other element of $S$ but $z$.
{{qed}}
Theorem: Group/Examples/Linear Functions

Let $G$ be the set of all real functions $\theta_{a, b}: \R \to \R$ defined as:
:$\forall x \in \R: \map {\theta_{a, b} } x = a x + b$
where $a, b \in \R$ such that $a \ne 0$.
The algebraic structure $\struct {G, \circ}$, where $\circ$ denotes composition of mappings, is a group.
$\struct {G, \circ}$ is specifically non-abelian.

Proof:
We verify the group axioms, in the following order (for convenience):
Theorem: Integer Multiples under Addition form Subgroup of Integers

Let $\struct {\Z, +}$ denote the additive group of integers.
Let $n \Z$ be the set of integer multiples of $n$.
Then $\struct {n \Z, +}$ is a subgroup of $\struct {\Z, +}$.
Hence $\struct {n \Z, +}$ can be justifiably referred to as the additive group of integer multiples.

Proof:
Clearly $0 \in n \Z$ so $n \Z \ne \O$.
Now suppose $x, y \in n \Z$.
Then $\exists r, s \in \Z: x = n r, y = n s$.
Also, $-y = - n s$.
So $x - y = n \paren {r - s}$.
As $r - s \in \Z$ it follows that $x - y \in n \Z$.
So by the One-Step Subgroup Test it follows that $\struct {n \Z, +}$ is a subgroup of the additive group of integers $\struct {\Z, +}$.
{{qed}}
Theorem: Subgroup Generated by Subgroup

Let $G$ be a group.
Let $H \le G$ be a subgroup of $G$.
Then:
:$H = \gen H$
where $\gen H$ denotes the subgroup generated by $H$.

Proof:
By definition of generated subgroup, $\gen H$ is the smallest subgroup of $H$ containing $H$.
Hence the result.
{{qed}}
Theorem: Group is Generated by Itself

Let $G$ be a group.
Then:
:$G = \gen G$
where $\gen G$ denotes the group generated by $G$.

Proof:
By definition of generated subgroup, $\gen G$ is the smallest subgroup of $G$ containing $G$.
Hence the result by Group is Subgroup of Itself.
{{qed}}
Theorem: Action of Inverse of Group Element

Let $\struct {G, \circ}$ be a group.
Let $S$ be a sets.
Let $*: G \times S \to S$ be a group action.
Then:
:$g * a = b \iff g^{-1} * b = a$

Proof:
{{begin-eqn}}
{{eqn | l = g * a
      | r = b
      | c = 
}}
{{eqn | ll= \leadsto
      | l = g^{-1} * \paren {g * a}
      | r = g^{-1} * b
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \paren {g^{-1} \circ g} * a
      | r = g^{-1} * b
      | c = {{GroupActionAxiom|2}}
}}
{{eqn | ll= \leadsto
      | l = e * a
      | r = g^{-1} * b
      | c = {{Group-axiom|3}}
}}
{{eqn | ll= \leadsto
      | l = a
      | r = g^{-1} * b
      | c = {{GroupActionAxiom|1}}
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn | l = g^{-1} * b
      | r = a
      | c = 
}}
{{eqn | ll= \leadsto
      | l = g * \paren {g^{-1} * b}
      | r = g * a
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \paren {g \circ g^{-1} } * b
      | r = g * a
      | c = {{GroupActionAxiom|2}}
}}
{{eqn | ll= \leadsto
      | l = e * b
      | r = g * a
      | c = {{Group-axiom|3}}
}}
{{eqn | ll= \leadsto
      | l = b
      | r = g * a
      | c = {{GroupActionAxiom|1}}
}}
{{end-eqn}}
{{qed}}