# Category:Improvements

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These are pages containing proofs that, although correct, are not the best they could be.

See the Improve template.

## Pages in category "Improvements"

The following 162 pages are in this category, out of 162 total.

### A

### B

### C

- Cardinal Equal to Collection of All Dominated Ordinals
- Characteristic of Ordered Integral Domain is Zero
- Characteristics of Traversable Graph
- Characterization of Lipschitz Continuity on Shift of Finite Type by Variations
- Definition:Chess
- Circle of Apollonius is Circle
- Circle of Apollonius is Circle/Proof 2
- Definition:Class Interval
- Definition:Classical Propositional Logic
- Closure of Irreducible Subspace is Irreducible
- Compact First-Countable Space is Sequentially Compact
- Compact Subspace of Linearly Ordered Space/Reverse Implication
- Compact Subspace of Linearly Ordered Space/Reverse Implication/Proof 2
- Compactness Theorem/Proof using Henkin Construction
- Composition of Mappings is Associative
- Condition for Ideal to be Total Ring
- Condition on Congruence Relations for Cancellable Monoid to be Group/Counterexample
- Conditions for Strong Minimum of Functional
- Definition:Connection on Manifold
- Definition:Consistent (Logic)/Set of Formulas/Propositional Logic
- Definition:Contravariant Hom Functor
- Definition:Convex Polygon/Definition 3
- Cosine of Sum/Proof 4
- Countable Union of Finite Sets is Countable
- Definition:Covariant Hom Functor
- Cycle does not Contain Subcycles

### D

- De Morgan's Laws (Predicate Logic)/Denial of Universality/Formulation 2/Forward Implication
- 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 Even Power of Sine x/Proof 1
- Definite Integral from 0 to Half Pi of Odd Power of Sine x
- Definite Integral from 0 to Half Pi of Odd Power of Sine x/Proof 1
- Determinant of Linear Operator is Well Defined
- Dimension of R-Module R is 1
- Dirichlet Integral/Proof 2
- Disjunction has no Inverse

### E

- Eigenvalues of Normal Operator have Orthogonal Eigenspaces
- Empty Set is Well-Ordered
- Equation for Perpendicular Bisector of Two Points
- Equivalence of Definitions of Normal Subgroup
- 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 Theorem for Planar Graphs
- Definition:Exact Sequence of Modules
- Existential Instantiation/Informal Statement
- Expansion Theorem for Determinants
- Exponential of Sum/Real Numbers/Corollary

### F

### G

### H

### I

- Image of Small Class under Mapping is Small
- Template:Improve
- Increasing Sequence in Ordered Set Terminates iff Maximal Element
- Increasing Sequence of Sets forms Nest
- Indexed Summation of Multiple of Mapping
- Intersection of Ordinals is Smallest
- Definition:Inverse Sine/Real/Arcsine
- Definition:Irreducible (Representation Theory)
- Definition:Irreducible (Representation Theory)/G-Module

### L

- Least Upper Bound Property/Proof 1
- Definition:Legendre Transform
- Lexicographic Product of Family of Ordered Sets is Ordered Set
- Limit Ordinal Equals its Union
- Linear Combination of Measures
- Liouville's Theorem (Complex Analysis)
- Liouville's Theorem (Complex Analysis)/Proof 1
- Logarithmic Integral as Non-Convergent Series

### M

### N

- Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite
- Necessary Condition for Integral Functional to have Extremum for given function/Dependent on N Functions
- Necessary Condition for Integral Functional to have Extremum for given Function/Dependent on Nth Derivative of Function
- Newton's Identities/Proof 1
- Non-Equivalence as Equivalence with Negation/Formulation 1/Forward Implication/Proof
- Norm on Vector Space is Quasinorm
- Definition:Number Base/Integers
- Number of Arrangements of n Objects of m Types
- Number which is Square and Cube Modulo 7

### O

- Open Sets of Double Pointed Topology/Corollary
- Order Isomorphism iff Strictly Increasing Surjection
- Order Topology equals Dual Order Topology
- Order Topology on Convex Subset is Subspace Topology
- Order-Extension Principle/Strict/Proof 2
- Ordinal Subset is Well-Ordered
- Orthocenter, Centroid and Circumcenter Coincide iff Triangle is Equilateral
- Orthonormal Subset of Hilbert Space Extends to Basis

### P

- P-Product Metrics on Real Vector Space are Topologically Equivalent
- P-Product Metrics on Real Vector Space are Topologically Equivalent/Inequality for General Case
- Primary Decomposition Theorem
- Prime Magic Square/Examples/Order 3/Smallest
- Primitive of x over Root of a x + b by Root of p x + q
- Product of Divisor Sum and Euler Phi Functions
- Products of 2-Digit Pairs which Reversed reveal Same Product
- Definition:Projection
- Projection from Product Category
- Definition:Proper Relational Structure
- Pullback of Quotient Group Isomorphism/Examples/C6 with A4

### R

- Rational Numbers with Denominators Coprime to Prime under Addition form Group
- Real Numbers of Type Rational a plus b root 2 form Field
- Reciprocal of Function of Bounded Variation Bounded away from Zero is of Bounded Variation
- 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

### S

- Scheffé's Lemma/Corollary
- Seifert-van Kampen Theorem
- Definition:Set of Literals
- Definition:Set Partition
- Shape of Cosecant Function
- Shape of Cosine Function
- Shape of Secant Function
- Sigma-Algebra Generated by Finite Partition is Finite Sub-Sigma-Algebra
- Sign of Composition of Permutations
- Sine and Cosine are Periodic on Reals/Corollary/Cosine
- Sine and Cosine are Periodic on Reals/Corollary/Sine
- Smallest Field is Field
- Solution to Legendre's Differential Equation
- Stopped Supermartingale is Supermartingale
- Strictly Increasing Sequence induces Partition
- Strictly Increasing Sequence of Natural Numbers
- Subband of Induced Operation is Set of Subbands
- Subgroup is Subset of Conjugate iff Normal
- Subgroup is Superset of Conjugate iff Normal
- Sufficient Conditions for Uncountability
- Sum of Indexed Suprema
- Sums of Squares of Diagonals of Order 3 Magic Square
- Supremum of Absolute Value of Difference equals Supremum of Difference