# File:Brain.png

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## File usage

The following 96 pages link to this file:

- Altitude, Median and Perpendicular Bisector Coincide iff Triangle is Isosceles
- Arcsin as an Integral/Lemma 2
- Area of Circle
- Area of Circle/Proof 3
- Banach Fixed-Point Theorem
- Bases of Finitely Generated Vector Space have Equal Cardinality
- Body under Constant Acceleration
- Cantor's Theorem (Strong Version)/Proof 2
- Characteristic of Ordered Integral Domain is Zero
- Compact First-Countable Space is Sequentially Compact
- Compact Subspace of Linearly Ordered Space
- Compact Subspace of Linearly Ordered Space/Reverse Implication
- Compact Subspace of Linearly Ordered Space/Reverse Implication/Proof 2
- Compactness Theorem
- Condition for Ideal to be Total Ring
- Conditions for Strong Minimum of Functional
- Continuum Property/Proof 1
- Cosine of Sum/Proof 4
- Countable Union of Finite Sets is Countable
- De Morgan's Laws imply Uniquely Complemented Lattice is Boolean Lattice
- Definite Integral from 0 to Half Pi of Even Power of Sine x
- Definite Integral from 0 to Half Pi of Odd Power of Sine x
- Diagonals of Rhombus Bisect Angles
- Disjunction has no Inverse
- Divergent Series/Examples/n over n^2 + i
- Eigenvalues of Normal Operator have Orthogonal Eigenspaces
- Equation for Perpendicular Bisector of Two Points
- Equivalence of Definitions of Transitive Closure (Set Theory)
- Equivalence of Subobjects is Equivalence
- Euler's Equation for Vanishing Variation is Invariant under Coordinate Transformations
- Euler's Number: Limit of Sequence implies Base of Logarithm
- Euler Polyhedron Formula
- Exponential of Sum/Real Numbers/Corollary
- Final Topology with respect to Mapping
- Finite Sequences in Set Form Acyclic Graph
- Floquet's Theorem/Proof 2
- Gaussian Integers form Euclidean Domain
- Gaussian Integers form Euclidean Domain/Proof 1
- General Variation of Integral Functional/Dependent on n Variables
- Image of Small Class under Mapping is Small
- Increasing Sequence in Ordered Set Terminates iff Maximal Element
- Indexed Summation of Multiple of Mapping
- Intersection of Ordinals is Smallest
- Inverse of Vandermonde Matrix
- Laplace Transform of Cosine/Proof 1
- Limit Point is Limit of Convergent Sequence/Proof 2
- Logarithmic Integral as Non-Convergent Series
- Mapping from Totally Ordered Set is Order Embedding iff Strictly Increasing/Reverse Implication
- Necessary Condition for Integral Functional to have Extremum for given Function/Dependent on N-th Derivative of Function
- Non-Equivalence as Equivalence with Negation/Formulation 1/Forward Implication/Proof
- Open Sets of Double Pointed Topology/Corollary
- Order-Extension Principle/Strict/Proof 2
- Order Topology equals Dual Order Topology
- Order Topology on Convex Subset is Subspace Topology
- Ordinal Subset is Well-Ordered
- Orthocenter, Centroid and Circumcenter Coincide iff Triangle is Equilateral
- P-Product Metrics on Real Vector Space are Topologically Equivalent
- P-Product Metrics on Real Vector Space are Topologically Equivalent/Inequality for General Case
- Perpendicular Bisector of Triangle is Altitude of Medial Triangle
- Primary Decomposition Theorem
- Primitive of Reciprocal of x squared plus a squared/Proof 3
- Primitive of x over Root of a x + b by Root of p x + q
- Product of Sigma and Euler Phi Functions
- Projection from Product Category
- Pullback of Quotient Group Isomorphism/Examples/C6 with A4
- Rational Numbers with Denominators Coprime to Prime under Addition form Group
- Recursively Defined Sequence/Examples/Term is Term of Index less 1 plus 6 times Term of Index less 2
- Relation between Two Ordinals
- Relationship between Transitive Closure Definitions
- Seifert-van Kampen Theorem
- Sign of Composition of Permutations
- Sine to Power of Odd Integer
- Solution to Legendre's Differential Equation
- Strictly Increasing Sequence induces Partition
- Strictly Increasing Sequence of Natural Numbers
- Subband of Induced Operation is Set of Subbands
- Subset Relation is Compatible with Subset Product
- Subset Relation is Compatible with Subset Product/Corollary 2
- Sufficient Conditions for Uncountability
- Sum of Indexed Suprema
- Sum of Sequence of Seventh Powers
- Sums of Squares of Diagonals of Order 3 Magic Square
- Supremum of Absolute Value of Difference equals Supremum of Difference
- Total Ordering on Quotient Field is Unique
- Volume of Cone is Third of Cylinder on Same Base and of Same Height
- Volume of Cone is Third of Cylinder on Same Base and of Same Height/Proof 2
- User:DingChao/Sandbox/Compactness Theorem
- Definition:Algebraic Element of Ring Extension
- Definition:Class (Descriptive Statistics)
- Definition:Contravariant Hom Functor
- Definition:Covariant Hom Functor
- Definition:Formal Semantics
- Definition:Hilbert Proof System
- Definition:Irreducible (Representation Theory)
- Definition:Irreducible (Representation Theory)/G-Module
- Definition:Magma of Sets