hcju/proofwikips huggingface dataset:
Equivalence of Formulations of Pasch's Axiom
The two forms of Pasch's Axiom in Tarski's Geometry are consistent. That is, the expressions: :$(1): \quad \forall a, b, c, p, q: \exists x: \mathsf B a p c \land \mathsf B b q c \implies \mathsf B p x b \land \mathsf B q x a$ and: :$(2): \quad \forall a, b, c, p, q: \exists x: \mathsf B a p c \land \mathsf B q c b \implies \mathsf B a x q \land \mathsf B b p x$ are logically equivalent.
Square-Summable Indexed Sets Closed Under Addition
Let $\family {a_i}_{i \mathop \in I}, \family {b_i}_{i \mathop \in I}$ be $I$-indexed families of real numbers. Let: :$\displaystyle \sum \set {a_i^2: i \in I} < \infty$ :$\displaystyle \sum \set {b_i^2: i \in I} < \infty$ where $\displaystyle \sum$ denotes the generalized sums. Then: :$\displaystyle \sum \set {\paren {a_i + b_i}^2: i \in I} < \infty$
Generalized Sum Preserves Inequality
Let $\left({a_i}\right)_{i \in I}, \left({b_i}\right)_{i \in I}$ be $I$-indexed families of positive real numbers. That is, let $a_i, b_i \in \R_{\ge 0}$ for all $i \in I$. Suppose that for all $i \in I$, $a_i \le b_i$. Furthermore, suppose that $\displaystyle \sum \left\{{ b_i: i \in I }\right\}$ converges. Then: : $\displaystyle \sum \left\{{ a_i: i \in I }\right\} \le \sum \left\{{ b_i: i \in I }\right\}$ In particular, $\displaystyle \sum \left\{{ a_i: i \in I }\right\}$ converges.
Generalized Sum is Linear
Let $\left({z_i}\right)_{i \in I}, \left({w_i}\right)_{i \in I}$ be $I$-indexed families of complex numbers. That is, let $z_i, w_i \in \C$ for all $i \in I$. Suppose that $\displaystyle \sum \left\{{ z_i: i \in I }\right\}, \sum \left\{{ w_i: i \in I }\right\}$ converge to $z, w \in \C$, respectively. Then: :$(1): \quad \displaystyle \sum \left\{{ z_i + w_i: i \in I }\right\}$ converges to $z+w$ :$(2): \quad \forall \lambda \in \C: \displaystyle \sum \left\{{ \lambda z_i: i \in I }\right\}$ converges to $\lambda z$
Min yields Infimum of Parameters
Let $\left({S, \preceq}\right)$ be a totally ordered set.  Let $x, y \in S$. Then: : $\min \left({x, y}\right) = \inf \left({\left\{{x, y}\right\}}\right)$ where: :$\min$ denotes the min operation :$\inf$ denotes the infimum.
Convergence of Generalized Sum of Complex Numbers
Let $\left({z_i}\right)_{i \mathop \in I}$ be an $I$-indexed family of complex numbers. That is, let $z_i \in \C$ for all $i \in I$. Denote by $\operatorname{Re} z_i$, resp. $\operatorname{Im} z_i$ the families of real, resp. imaginary parts of the family $z_i$. Then the following are equivalent: :$(1): \quad \displaystyle \sum \left\{{z_i : i \in I}\right\}$ converges to $z \in \C$ :$(2): \quad \displaystyle \sum \left\{{\operatorname{Re} z_i : i \in I}\right\}, \sum \left\{{\operatorname{Im} z_i : i \in I}\right\}$ converge to $\operatorname{Re} z, \operatorname{Im} z \in \R$, respectively
Lower and Upper Bounds for Sequences/Corollary
Let $\sequence {x_n}, \sequence {y_n}$ be sequences in $\R$. Let $x_n \to l, y_n \to m$ as $n \to \infty$. Suppose that for all $n \in \N$, $x_n \le y_n$. Then: :$l \le m$ that is: :$\displaystyle \lim_{n \mathop \to \infty} x_n \le \lim_{n \mathop \to \infty} y_n$ This is often phrased as: ''limits preserve inequalities''.
Bounded Linear Transformation Induces Bounded Sesquilinear Form
Let $H, K$ be Hilbert spaces over $\Bbb F \in \left\{{\R, \C}\right\}$. Let $A \in B \left({H, K}\right), B \in B \left({K, H}\right)$ be bounded linear transformations. Let $u, v: H \times K \to \Bbb F$ be defined by: :$u \left({h, k}\right) := \left\langle{Ah, k}\right\rangle_K$ :$v \left({h, k}\right) := \left\langle{h, Bk}\right\rangle_H$ Then $u$ and $v$ are bounded sesquilinear forms.
Classification of Bounded Sesquilinear Forms
Let $H, K$ be Hilbert spaces over $\Bbb F \in \left\{{\R, \C}\right\}$. Let $u: H \times K \to \Bbb F$ be a bounded sesquilinear form with bound $M$. Then there exist unique bounded linear transformations $A \in B \left({H, K}\right), B \in B \left({K, H}\right)$ such that: :$\forall h \in H, k \in K: u \left({h, k}\right) = \left\langle{Ah, k}\right\rangle_K = \left\langle{h, Bk}\right\rangle_H$ Furthermore, $\left\Vert{A}\right\Vert, \left\Vert{B}\right\Vert \le M$.
Isomorphism iff Inverse Equals Adjoint
Let $H, K$ be Hilbert spaces. Let $U \in B \struct {H, K}$ be a bounded linear transformation. Then the following are equivalent: :$(1): \quad U$ is an isomorphism :$(2): \quad U$ is invertible and $U^{-1} = U^*$, where $U^*$ denotes the adjoint of $U$.
Limit of Sequence is Limit of Real Function
Let $\sequence {a_n}$ be a real sequence. Let $f: x \mapsto \map f x$ be a real function. Suppose the limit: :$\displaystyle \lim_{x \mathop \to +\infty} \map f x$ exists. If for every $n$ in the domain of $\sequence {a_n}$: :$\map f n = a_n$ then: :$\displaystyle \lim_{n \mathop \to +\infty} \ a_n = \displaystyle \lim_{x \mathop \to +\infty} \map f x$
Equality of Cartesian Products
Let $A, B, C, D$ be nonempty sets. Then: : $A \times B = C \times D \iff A = C \land B = D$ where $\times$ denotes cartesian product.
Adjoining is Linear
Let $H, K$ be Hilbert spaces over $\Bbb F \in \left\{{\R, \C}\right\}$. Let $A, B \in B \left({H, K}\right)$ be bounded linear transformations. Then the operation of adjoining $^*$ satisfies, for all $\lambda \in \Bbb F$: :$(1): \qquad \left({\lambda A}\right)^* = \overline \lambda A^*$ :$(2): \qquad \left({A + B}\right)^* = A^* + B^*$ That is, $^*: B \left({H, K}\right) \to B \left({K, H}\right)$ is a linear transformation.
Adjoint of Composition
Let $H, K, L$ be Hilbert spaces. Let $A \in \map B {K, L}, B \in \map B {H, K}$ be bounded linear transformations. Then $\paren {A B}^* = B^* A^*$, where $^*$ denotes adjoining.
Double Adjoint is Itself
Let $H, K$ be Hilbert spaces. Let $A \in B \left({H, K}\right)$ be a bounded linear transformation. Then $A^{**} := \left({A^*}\right)^* = A$.
Identity of Points
Two points share the same position  they are the same points.
Successor of Omega
:$\omega + 1 = \set {0, 1, 2, ...; \omega}$ where $\omega$ is the minimal infinite successor set and $\omega + 1$ is the successor of $\omega$. Note the use of the semicolon; this is the notation for multipart infinite sets.
Equivalence of Versions of Axiom of Choice
The following formulations of the Axiom of Choice are equivalent:
Norm of Adjoint
Let $H, K$ be Hilbert spaces. Let $A \in \map B {H, K}$ be a bounded linear transformation. Then the norm of $A$ satisfies: :$\norm A^2 = \norm {A^*}^2 = \norm {A^* A}$ where $A^*$ denotes the adjoint of $A$.
Operator Self-Adjoint iff Inner Product Real
Let $H$ be a Hilbert space over $\C$. Let $A \in \map B H$ be a bounded linear operator. Then $A$ is self-adjoint : :$\forall h \in H: \innerprod {A h} h_H \mathop \in \R$
Norm of Self-Adjoint Operator
Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a self-adjoint operator. Then the norm of $A$ satisfies: :$\left\Vert{A}\right\Vert = \sup \left\{{ \left\vert{ \left\langle{Ah, h}\right\rangle_H }\right\vert: h \in H, \left\Vert{h}\right\Vert_H = 1 }\right\}$
Operator Zero iff Inner Product Zero
Let $H$ be a Hilbert space over $\C$. Let $A \in B \left({H}\right)$ be a bounded linear operator. Suppose that: :$\forall h \in H: \left\langle{Ah, h}\right\rangle_H = 0$ Then $A$ is the zero operator.
Linear Operator is Sum of Real and Imaginary Parts
Let $H$ be a Hilbert space over $\C$. Let $A \in B \left({H}\right)$ be a bounded linear operator. Let $B$ and $C$ be the real and imaginary parts of $A$, respectively. Then $A = B + iC$.
Characterization of Normal Operators
Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a bounded linear operator. Then the following are equivalent: :$(1): \qquad AA^* = A^*A$, i.e., $A$ is normal :$(2): \qquad \forall h \in H: \left\Vert{Ah}\right\Vert_H = \left\Vert{A^*h}\right\Vert_H$ If $H$ is a Hilbert space over $\C$, these are also equivalent to: :$(3): \qquad \operatorname{Re} \left({A}\right) \operatorname{Im} \left({A}\right) = \operatorname{Im} \left({A}\right) \operatorname{Re} \left({A}\right)$, i.e., the real and imaginary parts of $A$ commute
Isometry iff Adjoint is Left-Inverse
Let $H, K$ be Hilbert spaces. Let $A \in B \left({H, K}\right)$ be a bounded linear transformation. Then $A$ is an isometry iff: :$A^*A = I_H$ where $A^*$ denotes the adjoint of $A$, and $I_H$ the identity operator on $H$.
Characterization of Unitary Operators
Let $H$ be a Hilbert space. Let $A \in \map B H$ be a bounded linear operator. Then the following are equivalent: :$(1): \quad A$ is a unitary operator :$(2): \quad A^* A = A A^* = I$, where $A^*$ denotes the adjoint of $A$, and $I$ denotes the identity operator :$(3): \quad A$ is a normal isometry
Kernel of Linear Transformation is Orthocomplement of Range of Adjoint
Let $H, K$ be Hilbert spaces. Let $A \in B \struct {H, K}$ be a bounded linear transformation. Then $\ker A = \paren {\Img {A^*} }^\perp$, where: :$A^*$ denotes the adjoint of $A$ :$\ker A$ is the kernel of $A$ :$\Img {A^*}$ is the image of $A^*$ :$\perp$ signifies orthocomplementation
Relation Contains Mapping is Equivalent to AoC
Let $S$ and $T$ be sets. Let $\RR \subseteq S \times T$ be a relation on $S \times T$. Then: :there exists a mapping $f \subseteq \RR$ whose domain is the same as the preimage of $\RR$  :the axiom of choice holds.
Existence of Set with Singleton Intersections with Disjoint Collection
Let $\mathcal C$ be a set of sets all of which are pairwise disjoint. Then: : there exists a set $A$ such that $\forall S \in \mathcal C: A \cap S$ is a singleton  : the axiom of choice holds.
Complementary Idempotent is Idempotent
Let $H$ be a Hilbert space. Let $A$ be an idempotent operator. Then the complementary idempotent $I - A$ is also idempotent.
Range of Idempotent is Kernel of Complementary Idempotent
Let $H$ be a Hilbert space. Let $A$ be an idempotent operator. Then $\Rng A = \map \ker {I - A}$.
Range and Kernel of Idempotent are Algebraically Complementary
Let $H$ be a Hilbert space. Let $A$ be an idempotent operator. Then $\ker A$ and $\Rng A$ are algebraically complementary, that is: :$\ker A \cap \Rng A = \left({0}\right)$, the zero subspace :$\ker A + \Rng A = H$, where $+$ signifies setwise addition. {{explain|Determine whether range means image or codomain, and replace appropriately}}
Natural Logarithm Function is Continuous
The natural logarithm function is continuous.
Characterization of Projections
Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be an idempotent operator. Then the following are equivalent: :$(1): \qquad A$ is a projection :$(2): \qquad A$ is the orthogonal projection onto $\operatorname{ran} A$ :$(3): \qquad \left\Vert{A}\right\Vert = 1$, where $\left\Vert{\cdot}\right\Vert$ is the norm on bounded linear operators. :$(4): \qquad A$ is self-adjoint :$(5): \qquad A$ is normal :$(6): \qquad \forall h \in H: \left\langle{Ah, h}\right\rangle_H \ge 0$
Set of Linear Subspaces is Closed under Intersection
Let $\struct {V, +, \circ}_K$ be a $K$-vector space. Let $\family {M_i}_{i \mathop \in I}$ be an $I$-indexed family of subspaces of $V$. Then $M := \ds \bigcap_{i \mathop \in I} M_i$ is also a subspace of $V$.
Closed Linear Subspaces Closed under Intersection
Let $V$ be a topological vector space. Let $\family {M_i}_{i \mathop \in I}$ be an $I$-indexed family of closed linear subspaces of $V$. Then $M := \ds \bigcap_{i \mathop \in I} M_i$ is also a closed linear subspace of $V$.
Orthogonal Difference is Closed Linear Subspace
Let $H$ be a Hilbert space. Let $M, N$ be closed linear subspaces of $H$. Then the orthogonal difference $M \ominus N$ is also a closed linear subspace of $H$.
Closed Linear Subspaces Closed under Setwise Addition
Let $H$ be a Hilbert space. Let $M, N$ be closed linear subspaces of $H$. Then $M + N$ is also a closed linear subspace of $H$, where $+$ denotes setwise addition.
Linear Subspaces Closed under Setwise Addition
Let $V$ be a $K$-vector space. Let $M, N$ be linear subspaces of $V$. Then $L := M + N$ is also a linear subspace of $V$, where $+$ denotes setwise addition.
Complementary Projection is Projection
Let $H$ be a Hilbert space. Let $A$ be a projection. Then the complementary projection $I - A$ is also a projection.
Direct Sum of Subspace and Orthocomplement
Let $H$ be a Hilbert space. Let $M$ be a closed linear subspace of $H$. Denote by $M^\perp$ its orthocomplement. Then the direct sum $M \oplus M^\perp$ is isomorphic to $H$.
Characterization of Invariant Subspaces
Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a bounded linear operator. Let $\begin{pmatrix}  W & X \\  Y & Z \end{pmatrix}$ be the matrix notation for $A$ with respect to $M$. Let $M$ be a closed linear subspace of $H$; denote by $P$ the orthogonal projection on $M$. Then the following three statements are equivalent: :$(1): \qquad M$ is an invariant subspace for $A$ :$(2): \qquad PAP = AP$ :$(3): \qquad Y = 0$
Characterization of Reducing Subspaces
Let $\HH$ be a Hilbert space. Let $A \in \map B \HH$ be a bounded linear operator. Let $M$ be a closed linear subspace of $\HH$. Let $P$ denote the orthogonal projection on $M$. Let $\begin{pmatrix}  W & X \\  Y & Z \end{pmatrix}$ be the matrix notation for $A$ with respect to $M$. Then the following four statements are equivalent: :$(1): \quad M$ is a reducing subspace for $A$ :$(2): \quad P A = A P$ :$(3): \quad X = Y = 0$ :$(4): \quad M$ is an invariant subspace for both $A$ and its adjoint $A^*$
Subset of Toset is Toset
Let $\left({S, \preceq}\right)$ be a totally ordered set. Let $T \subseteq S$. Then $\left({T, \preceq \restriction_T}\right)$ is also a totally ordered set. In the above, $\preceq \restriction_T$ denotes the restriction of $\preceq$ to $T$.
Bretschneider's Formula
Let $ABCD$ be a general quadrilateral. Then the area $\mathcal A$ of $ABCD$ is given by: :$\mathcal A = \sqrt{\left({s - a}\right) \left({s - b}\right) \left({s - c}\right) \left({s - d}\right) - a b c d \cos^2 \left({\dfrac {\alpha + \gamma} 2}\right)}$ where: : $a, b, c, d$ are the lengths of the sides of the quadrilateral : $s = \dfrac {a + b + c + d} 2$ is the semiperimeter : $\alpha$ and $\gamma$ are opposite angles.
Transfinite Induction/Principle 1
Let $\On$ denote the class of all ordinals. Let $A$ denote a class. Suppose that: :For all elements $x$ of $\On$, if $x$ is a subset of $A$, then $x$ is an element of $A$. Then $\On \subseteq A$.
Limit of Functions that Agree
Let $f$ and $g$ be real functions. Let $f$ and $g$ agree for all $x$ in a deleted neighborhood of $c$. Let the limit: :$\displaystyle \lim_{x \to c} \ f \left({x}\right)$ exist. Then the limit: :$\displaystyle \lim_{x \to c} \ g \left({x}\right)$ also exists, and: :$\displaystyle \lim_{x \to c} \ f \left({x}\right) =  \lim_{x \to c} \ g \left({x}\right)$
Ordering of Reciprocals
Let $x, y \in \R$ be real numbers such that $x, y \in \openint 0 \to$ or $x, y \in \openint \gets 0$ Then: :$x \le y \iff \dfrac 1 y \le \dfrac 1 x$
Comparison Test for Divergence
Let $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ be a divergent series of positive real numbers. Let $\sequence {a_n}$ be a sequence in $\R$. Let: :$\forall n \in \N_{>0}: b_n \le a_n$ Then the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ diverges.
Limit at Infinity of Real Identity Function
Let $I_\R: \R \to \R$ be the identity function on $\R$. Then: :$(1): \quad \displaystyle \lim_{x \mathop \to +\infty} \map {I_\R} x = +\infty$ :$(2): \quad \displaystyle \lim_{x \mathop \to -\infty} \map {I_\R} x = -\infty$
Push Theorem
Let $f$ be a real function which is continuous on the open interval $\openint a \to$, $a \in \R$, such that: :$\displaystyle \lim_{x \mathop \to +\infty} \map f x = +\infty$ Let $g$ be a real function defined on some open interval $\openint b \to$ such that, for sufficiently large $x$: :$\map g x > \map f x$ Then: :$\displaystyle \lim_{x \mathop \to +\infty} \map g x = +\infty$
Orthocomplement Reverses Subset
Let $H$ be a Hilbert space. Let $A, B$ be subsets of $H$, and let $A \subseteq B$. Then $B^\perp \subseteq A^\perp$, where $\perp$ signifies orthocomplementation.
Limit at Infinity of x^n
Let $x \mapsto x^n$, $n \in \R$ be a real function which is continuous on the open interval $\openint 1 {+\infty}$. Let $n > 0$. Then $x^n \to +\infty$ as $x \to +\infty$.
Monotonicity of Real Sequences
Let $\sequence {a_n}: \mathbb D \to \R$ be a real sequence, where $\mathbb D$ is a subset of $\N$. Let $\Bbb X$ be a real interval such that $\Bbb D \subseteq \Bbb X$. Let $f: \Bbb X \to \R, x \mapsto \map f x$ be a differentiable real function. Suppose that for every $n \in \mathbb D$: :$\map f n = a_n$ Then: :If $\forall x \in \Bbb X: D_x \map f x \ge 0$, $\sequence {a_n}$ is increasing :If $\forall x \in \Bbb X: D_x \map f x >  0$, $\sequence {a_n}$ is strictly increasing :If $\forall x \in \Bbb X: D_x \map f x \le 0$, $\sequence {a_n}$ is decreasing :If $\forall x \in \Bbb X: D_x \map f x < 0$, $\sequence {a_n}$ is strictly decreasing where $D_x$ denotes differentiation {{WRT|Differentiation}} $x$.
Hausdorff Maximal Principle is equivalent to Axiom of Choice
Every ordered set has a maximal chain  the axiom of choice holds.
Young's Inequality for Products
Let $p, q \in \R_{> 0}$ be strictly positive real numbers such that: : $\dfrac 1 p + \dfrac 1 q = 1$ Then, for any $a, b \in \R_{\ge 0}$: : $a b \le \dfrac {a^p} p + \dfrac{b^q} q$ Equality occurs : :$b = a^{p-1}$.
Restriction of Monotone Function is Monotone
The restriction of a monotone mapping is monotone.
Taylor's Theorem/One Variable/Statement of Theorem
Let $f$ be a real function which is: :of differentiability class $C^n$ on the closed interval $\closedint a x$ and: :at least $n + 1$ times differentiable on the open interval $\openint a x$. Then: {{begin-eqn}} {{eqn | l = \map f x       | r = \frac 1 {0!} \map f a       | c = }} {{eqn | o =       | ro= +       | r = \frac 1 {1!} \paren {x - a} \map {f'} a       | c =  }} {{eqn | o =       | ro= +       | r = \frac 1 {2!} \paren {x - a}^2 \map {f''} a       | c =  }} {{eqn | o =       | ro= +       | r = \cdots       | c =  }} {{eqn | o =       | ro= +       | r = \frac 1 {n!} \paren {x - a}^n \map {f^{\paren n} } a       | c =  }} {{eqn | o =       | ro= +       | r = R_n       | c =  }} {{end-eqn}} where $R_n$ (sometimes denoted $E_n$) is known as the '''error term''' or '''remainder''', and can be presented in one of $2$ forms: ;Lagrange Form :$R_n = \dfrac {\map {f^{\paren {n + 1} } } \xi} {\paren {n + 1}!} \paren {x - a}^{n + 1}$ for some $\xi \in \openint a x$. ;Cauchy Form :$R_n = \dfrac {\map {f^{\paren {n + 1} } } \xi \paren {x - \xi}^n} {n!} \paren {x - a}$ for some $\xi \in \openint a x$.
Elements of Minimal Infinite Successor Set are Well-Ordered
Let $\omega$ be the minimal infinite successor set. Let $a \in \omega$. Then $a$ is well-ordered by $\subseteq$.
Finite Lexicographic Order on Well-Ordered Sets is Well-Ordering
Let $S$ be a set which is well-ordered by $\preceq$. Let $\preccurlyeq$ be the lexicographic order on the set $T_n$ of all ordered $n$-tuples of $S$: :$\left({x_1, x_2, \ldots, x_n}\right) \prec \left({y_1, y_2, \ldots, y_n}\right)$ : ::$\exists k: 1 \le k \le n$ such that $\forall 1 \le j < k: x_j = y_j$ but $x_k \prec y_k$ in $S$. {{explain|Disentangle this.}} Then for a given $n \in \N_{>0}$, $\preccurlyeq$ is a well-ordering on $T_n$.
Infinite Lexicographic Order on Well-Ordered Sets is not Well-Ordering
Let $\struct {S, \preceq}$ be a well-ordered set. Let $\preccurlyeq$ be the lexicographic order on the set $T$ of all ordered tuples of $S$: :$\tuple {x_1, x_2, \ldots, x_m} \prec \tuple {y_1, y_2, \ldots, y_n}$ : ::$\exists k: 1 \le k \le \map \min {m, n}$ such that $\forall 1 \le j < k: x_j = y_j$ but $x_k \prec y_k$ in $S$ :or: ::$m < n$ and $\forall 1 \le j < m: x_j = y_j$. Then $\preccurlyeq$ is '''not''' a well-ordering on $T$.
Minimal Infinite Successor Set is Well-Ordered
Let $\omega$ be the minimal infinite successor set. Then $\omega$ is well-ordered by $\subseteq$.
Transfinite Induction
=== Principle 1 === {{:Transfinite Induction/Principle 1}} === Schema 1 === {{:Transfinite Induction/Schema 1}} === Principle 2 === {{:Transfinite Induction/Principle 2}} === Schema 2 === {{:Transfinite Induction/Schema 2}}
Transfinite Recursion/Theorem 1
Let $G$ be a mapping. {{explain|What are the domain and range of $G$?}} Let $K$ be a class of mappings $f$ that satisfy: :the domain of $f$ is some ordinal $y$ :$\forall x \in y: f \left({x}\right) = G \left({f {\restriction_x} }\right)$ where $f {\restriction_x}$ denotes the restriction of $f$ to $x$. Let $F = \bigcup K$, the union of $K$. Then: :$(1): \quad F$ is a mapping with domain $\operatorname{On}$ :$(2): \quad \forall x \in \operatorname{On}: F \left({x}\right) = G \left({F {\restriction_x} }\right)$ :$(3): \quad F$ is unique. That is, if another mapping $A$ has the above two properties, then $A = F$.
Squeeze Theorem for Absolutely Convergent Series
Let $\displaystyle \sum \size {a_n}$ be an absolutely convergent series in $\R$. Suppose that: :$\displaystyle -\sum \size {a_n} = \sum \size {a_n}$ Then $\displaystyle \sum a_n$ equals the above two series.
N Choose k is not greater than n^k
:$\forall n \in \Z, k \in \Z: 1 \le k \le n: \dbinom n k < n^k$  where $\dbinom n k$ is a binomial coefficient. Equality holds when $k = 0$ and $k = 1$.
Absolutely Convergent Series is Convergent/Real Numbers
Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be an absolutely convergent series in $\R$. Then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is convergent.
Definition:Natural Numbers/Zermelo Construction
The natural numbers $\N = \set {0, 1, 2, 3, \ldots}$ can be defined as a series of subsets: :$0 := \O = \set {}$ :$1 := \set 0 = \set \O$ :$2 := \set 1 = \set {\set \O}$ :$3 := \set 2 = \set {\set {\set \O} }$ ::::$\vdots$ Thus the natural number $n$ consists of $\O$ enclosed in $n$ pairs of braces.
Set is Element of its Power Set
A set is an element of its power set: :$S \in \powerset S$
Power Set of Empty Set
The power set of the empty set $\O$ is the set $\set \O$.
Symmetric Difference is Subset of Union
The symmetric difference of two sets is a subset of their union: :$S * T \subseteq S \cup T$
Sum of Projections/General Case
Let $H$ be a Hilbert space. Let $\left({M_i}\right)_{i \in I}$ be an $I$-indexed set of closed linear subspaces of $H$. Let $M_i$ and $M_j$ be orthogonal whenever $i \ne j$. Denote, for each $i \in I$, by $P_i$ the orthogonal projection onto $M_i$. Denote by $P$ the orthogonal projection onto the closed linear span $\vee \left\{{M_i: i \in I}\right\}$ of the $M_i$. Then for all $h \in H$, $\displaystyle \sum \left\{{P_i h: i \in I}\right\} = Ph$, where $\displaystyle \sum$ denotes a generalized sum.
Product of Projections
Let $H$ be a Hilbert space. Let $P, Q$ be projections. Then the following are equivalent: :$(1): \quad PQ$ is a projection :$(2): \quad PQ = QP$ :$(3): \quad P + Q - PQ$ is a projection {{MissingLinks|Provide proper linking to the def of addition and multiplication of operators}}
Difference of Projections
Let $H$ be a Hilbert space. Let $P, Q$ be projections. Then the following are equivalent: :$(1): \quad P - Q$ is a projection :$(2): \quad P Q = Q$ :$(3): \quad Q P = Q$ {{MissingLinks|Provide proper linking to the def of addition of operators}}
Union of Countable Sets of Sets
Let $\mathcal A$ and $\mathcal B$ be countable sets of sets. Then: :$\left\{{A \cup B: A \in \mathcal A, B \in \mathcal B}\right\}$ is also countable.
Set of Finite Subsets of Countable Set is Countable
Let $A$ be a countable set. Then the set of finite subsets of $A$ is countable.
Cantor's Theorem (Strong Version)
Let $S$ be a set. Let $\map {\PP^n} S$ be defined recursively by: :$\map {\PP^n} S = \begin{cases} S & : n = 0 \\ \powerset {\map {\PP^{n - 1} } S} & : n > 0 \end{cases}$ where $\powerset S$ denotes the power set of $S$. Then $S$ is not equivalent to $\map {\PP^n} S$ for any $n > 0$.
Ordering Induced by Injection is Ordering
Let $\left({T, \le}\right)$ be an ordered set, and let $S$ be a set. Let $f: S \to T$ be an injection. Then $\le_f$, the ordering induced by $f$, is an ordering.
Injection Induces Total Ordering
Let $\left({T, \le}\right)$ be a totally ordered set, and let $S$ be a set. Let $f: S \to T$ be an injection. Then $\le_f$, the ordering induced by $f$, is a total ordering.