nyuuzyou/wikis huggingface dataset:
Real Function of Two Variables/Examples

== Examples of Real Functions of Two Variables ==
=== Volume of Right Circular Cylinder ===
=== Example: $y \sqrt {1 - x^2}$ ===
=== Example: $\dfrac {\sqrt {1 - y^2} } {\sqrt {1 - x^2} }$ ===
=== Example: $x + y$ ===
=== Example: $\sqrt {x^2 + y^2 - 25}$ ===
=== Example: $\sqrt {-\paren {x^2 + y^2} }$ ===
=== Example: $\sqrt {-\paren {x^2 + y^2 + 1} }$ ===
=== Arbitrary Example ===
Category:Examples of Real Functions
Real Function of Two Variables/Examples/Arbitrary Example 1

== Example of Real Function of Two Variables ==
The real-valued function $f: \R^2 \to \R$:
:$\map f {x_1, x_2} = \sin x_1 + x_1 \cos x_2$
is an example of a real function of two wariables.
== Sources ==
* : Chapter $1$: Real Numbers and Functions of a Real Variable: $1.3$ Functions of a Real Variable: $\text {(c)}$ Functions of More than One Variable $(2)$
Category:Examples of Real Multivariable Functions
Real Function of Two Variables/Examples/Root of 1 minus y^2 over Root of 1 minus x^2

== Examples of Real Functions of Two Variables ==
Let $z$ denote the function defined as:
:$z = \dfrac {\sqrt {1 - y^2} } {\sqrt {1 - x^2} }$
The domain of $z$ is:
:$\Dom z = \openint {-1} 1 \times \closedint {-1} 1$
== Proof ==
The domain of $z$ is given implicitly and conventionally.
What is meant is:
:$z: S \to \R$ is the function defined on the largest possible subset $S$ of $\R^2$ such that:
:$\forall \tuple {x, y} \in S: \map z {x, y} = \dfrac {\sqrt {1 - y^2} } {\sqrt {1 - x^2} }$
From Domain of Real Square Root Function, in order for the real square root function to be defined, its argument must be non-negative.
Hence for $z$ to be defined, it is necessary for:
Similarly:
:$\size x \le 1$
But because $\sqrt {1 - x^2}$ is the denominator of the , it cannot be zero.
So it follows that for $z$ to be defined, it must be the case that:
:$-1 \le y \le 1$
and:
:$-1 < x < 1$
Hence the result.
== Sources ==
* : Chapter $1$: Basic Concepts: Lesson $2 \text C$: Function of Two Independent Variables: Example $2.62: 1$
Category:Examples of Real Functions
Real Function of Two Variables/Examples/Root of Minus (x^2 + y^2)

== Examples of Real Functions of Two Variables ==
Let $z$ denote the function defined as:
:$z = \sqrt {-\paren {x^2 + y^2} }$
The domain of $z$ is:
:$\Dom z = \tuple {0, 0}$
That is, just one single point, the origin of the Cartesian plane
== Proof ==
The domain of $z$ is given implicitly and conventionally.
What is meant is:
:$z: S \to \R$ is the function defined on the largest possible subset $S$ of $\R^2$ such that:
:$\forall \tuple {x, y} \in S: \map z {x, y} = \sqrt {-\paren {x^2 + y^2} }$
From Domain of Real Square Root Function, in order for the real square root function to be defined, its argument must be non-negative.
Hence for $z$ to be defined, it is necessary for:
But:
:$\forall x, y \in \R: x^2 + y^2 \ge 0$
Hence the only point which fulfils the condition is $\tuple {0, 0}$.
Hence the result.
== Sources ==
* : Chapter $1$: Basic Concepts: Lesson $2 \text C$: Function of Two Independent Variables: Example $2.62: 4$
Category:Examples of Real Functions
Real Function of Two Variables/Examples/Root of Minus (x^2 + y^2 + 1)

== Examples of Real Functions of Two Variables ==
Let $z$ denote the function defined as:
:$z = \sqrt {-\paren {x^2 + y^2 + 1} }$
The domain of $z$ is:
:$\Dom z = \O$
That is, there are no points of the Cartesian plane for which $z$ is defined.
== Proof ==
The domain of $z$ is given implicitly and conventionally.
What is meant is:
:$z: S \to \R$ is the function defined on the largest possible subset $S$ of $\R^2$ such that:
:$\forall \tuple {x, y} \in S: \map z {x, y} = \sqrt {-\paren {x^2 + y^2 + 1} }$
From Domain of Real Square Root Function, in order for the real square root function to be defined, its argument must be non-negative.
Hence for $z$ to be defined, it is necessary for:
But:
:$\forall x, y \in \R: x^2 + y^2 \ge 0$
Hence the result.
== Sources ==
* : Chapter $1$: Basic Concepts: Lesson $2 \text C$: Function of Two Independent Variables: Example $2.62: 5$
Category:Examples of Real Functions
Real Function of Two Variables/Examples/Root of x^2 + y^2 - 25

== Examples of Real Functions of Two Variables ==
Let $z$ denote the function defined as:
:$z = \sqrt {x^2 + y^2 - 25}$
The domain of $z$ is:
:$\Dom z = C$
where $C$ consists of the set of points outside and on the circumference of the circle of radius $5$ whose center is at $\tuple {0, 0}$ in the Cartesian plane.
== Proof ==
The domain of $z$ is given implicitly and conventionally.
What is meant is:
:$z: S \to \R$ is the function defined on the largest possible subset $S$ of $\R^2$ such that:
:$\forall \tuple {x, y} \in S: \map z {x, y} = \sqrt {x^2 + y^2 - 25}$
From Domain of Real Square Root Function, in order for the real square root function to be defined, its argument must be non-negative.
Hence for $z$ to be defined, it is necessary for:
From Equation of Circle center Origin, $x^2 + y^2 = 25$ is the equation for the circle of radius $5$ whose center is at $\tuple {0, 0}$ in the Cartesian plane.
Points inside this circle correspond are such that $x^2 + y^2 < 25$.
Hence the domain of $z$ is the set of points consisting of the exterior of that circle and the points on its circumference.
Hence the result.
== Sources ==
* : Chapter $1$: Basic Concepts: Lesson $2 \text C$: Function of Two Independent Variables: Example $2.62: 3$
Category:Examples of Real Functions
Real Function of Two Variables/Examples/Volume of Right Circular Cylinder

== Examples of Real Functions of Two Variables ==
Let $C$ denote be a right circular cylinder.
Then the volume $V$ of $C$ is a function of:
:the radius $r$ of the base of $C$
:the height $h$ of $C$
such that:
:$V = \pi r^2 h$
== Sources ==
* 
* 
Category:Examples of Real Functions
Real Function of Two Variables/Examples/x + y

== Examples of Real Functions of Two Variables ==
Let $z$ denote the function defined as:
:$z = x + y$
The domain of $z$ is:
:$\Dom z = \R \times \R$
== Proof ==
The domain of $z$ is given implicitly and conventionally.
What is meant is:
:$z: S \to \R$ is the function defined on the largest possible subset $S$ of $\R^2$ such that:
:$\forall \tuple {x, y} \in S: \map z {x, y} = x + y$
There are no restrictions on either $x$ or $y$ for $\map z {x, y}$ to be defined.
Hence the domain of $z$ is the entire Cartesian plane.
== Sources ==
* : Chapter $1$: Basic Concepts: Lesson $2 \text C$: Function of Two Independent Variables: Example $2.62: 2$
Category:Examples of Real Functions
Real Function of Two Variables/Examples/y by Root of 1 minus x^2

== Examples of Real Functions of Two Variables ==
Let $z$ denote the function defined as:
:$z = y \sqrt {1 - x^2}$
The domain of $z$ is:
:$\Dom z = \closedint {-1} 1 \times \R$
== Proof ==
The domain of $z$ is given implicitly and conventionally.
What is meant is:
:$z: S \to \R$ is the function defined on the largest possible subset $S$ of $\R^2$ such that:
:$\forall \tuple {x, y} \in S: \map z {x, y} = y \sqrt {1 - x^2}$
From Domain of Real Square Root Function, in order for the real square root function to be defined, its argument must be non-negative.
Hence for $z$ to be defined, it is necessary for:
So for $z$ to be defined, it must be that $-1 \le x \le 1$.
There are no such restrictions on the value of $y$.
Hence the result.
== Sources ==
* : Chapter $1$: Basic Concepts: Lesson $2 \text C$: Function of Two Independent Variables
Category:Examples of Real Functions
Real Function of Two Variables/Substitution for y/Examples/x^2 + xy^2 + 5y + 3

== Examples of Substitution for $y$ in Real Function of Two Variables ==
Let $\map f {x, y}$ be the real function of $2$ variables defined as:
:$\forall \tuple {x, y} \in \R^2: \map f {x, y} := x^2 + x y^2 + 5 y + 3$
==== Substituting $2$ for $y$ ====
==== Substituting $a$ for $y$ ====
==== Substituting $\map g x$ for $y$ ====
Category:Examples of Real Functions
Real Function of Two Variables/Substitution for y/Examples/x^2 + xy^2 + 5y + 3/2

== Example of Substitution for $y$ in Real Function of Two Variables ==
Let $\map f {x, y}$ be the real function of $2$ variables defined as:
:$\forall \tuple {x, y} \in \R^2: \map f {x, y} := x^2 + x y^2 + 5 y + 3$
Let $2$ be substituted for $y$ in $\map f {x, y}$.
Then:
== Sources ==
* : Chapter $1$: Basic Concepts: Lesson $2 \text C$: Function of Two Independent Variables: Example $2.66 \ \text {(a)}$
Category:Examples of Real Functions
Real Function of Two Variables/Substitution for y/Examples/x^2 + xy^2 + 5y + 3/a

== Example of Substitution for $y$ in Real Function of Two Variables ==
Let $\map f {x, y}$ be the real function of $2$ variables defined as:
:$\forall \tuple {x, y} \in \R^2: \map f {x, y} := x^2 + x y^2 + 5 y + 3$
Let $a$ be substituted for $y$ in $\map f {x, y}$.
Then:
== Sources ==
* : Chapter $1$: Basic Concepts: Lesson $2 \text C$: Function of Two Independent Variables: Example $2.66 \ \text {(b)}$
Category:Examples of Real Functions
Real Function of Two Variables/Substitution for y/Examples/x^2 + xy^2 + 5y + 3/g (x)

== Example of Substitution for $y$ in Real Function of Two Variables ==
Let $\map f {x, y}$ be the real function of $2$ variables defined as:
:$\forall \tuple {x, y} \in \R^2: \map f {x, y} := x^2 + x y^2 + 5 y + 3$
Let $\map g x$ be substituted for $y$ in $\map f {x, y}$, where $\map g x$ is a real function defined on all $\R$.
Then:
== Sources ==
* : Chapter $1$: Basic Concepts: Lesson $2 \text C$: Function of Two Independent Variables: Example $2.66 \ \text {(c)}$
Category:Examples of Real Functions
Real Function of Two Variables/Substitution for y/Examples/x + y

== Examples of Substitution for $y$ in Real Function of Two Variables ==
Let $\map f {x, y}$ be the real function of $2$ variables defined on the domain $S \times T$ as:
:$\forall \tuple {x, y} \in S \times T: \map f {x, y} := x + y$
where $S$ and $T$ are the closed real intervals:
==== Substituting $\dfrac 1 2$ for $y$ ====
==== Substituting $3$ for $y$ ====
Category:Examples of Real Functions
Real Function of Two Variables/Substitution for y/Examples/x + y/1 over 2

== Examples of Substitution for $y$ in Real Function of Two Variables ==
Let $\map f {x, y}$ be the real function of $2$ variables defined on the domain $S \times T$ as:
:$\forall \tuple {x, y} \in S \times T: \map f {x, y} := x + y$
where $S$ and $T$ are the closed real intervals:
Let $\dfrac 1 2$ be substituted for $y$ in $\map f {x, y}$.
Then:
== Sources ==
* : Chapter $1$: Basic Concepts: Lesson $2 \text C$: Function of Two Independent Variables: Example $2.67 \ \text {(a)}$
Category:Examples of Real Functions
Real Function of Two Variables/Substitution for y/Examples/x + y/3

== Examples of Substitution for $y$ in Real Function of Two Variables ==
Let $\map f {x, y}$ be the real function of $2$ variables defined on the domain $S \times T$ as:
:$\forall \tuple {x, y} \in S \times T: \map f {x, y} := x + y$
where $S$ and $T$ are the closed real intervals:
Let $3$ be substituted for $y$ in $\map f {x, y}$.
Then $\map f {x, 3}$ is undefined, as $3 \notin \closedint 1 2$.
== Sources ==
* : Chapter $1$: Basic Concepts: Lesson $2 \text C$: Function of Two Independent Variables: Example $2.67 \ \text {(b)}$
Category:Examples of Real Functions
Real Function of Two Variables represents Surface in Cartesian 3-Space

== Theorem ==
Let $S$ and $T$ be subsets of the set of real numbers $\R$.
Let $f: S \times T \to \R$ be a real function of two variables.
Then the locus of $f$ describes a surface embedded in the Cartesian space $\R^3$.
== Proof ==
== Sources ==
* : $1$ Partial Differentiation: $\S 1$. Introduction
Category:Real Functions
Real Function with Negative Derivative is Decreasing

== Theorem ==
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.
Let:
:$\forall x \in \openint a b: \map {f'} x \le 0$
Then $f$ is decreasing on $\closedint a b$.
== Proof ==
Let $c, d \in \closedint a b$ such that $c < d$.
Then $f$ satisfies the conditions of the Mean Value Theorem on $\closedint c d$.
Hence:
:$\exists \xi \in \openint c d: \map {f'} \xi = \dfrac {\map f d - \map f c} {d - c}$
Let $f$ be such that:
:$\forall x \in \openint a b: \map {f'} x \le 0$
Then:
:$\map {f'} \xi \le 0$
and hence:
:$\map f d \le \map f c$
Thus $f$ is decreasing on $\closedint a b$.
== Sources ==
* : $\S 12.7 \ \text {(ii)}$
* 
* 
Category:Decreasing Real Functions
Category:Real Functions
Category:Differential Calculus
Real Function with Positive Derivative is Increasing

== Theorem ==
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.
Let:
:$\forall x \in \openint a b: \map {f'} x \ge 0$
Then $f$ is increasing on $\closedint a b$.
== Proof ==
Let $c, d \in \closedint a b$ such that $c < d$.
Then $f$ satisfies the conditions of the Mean Value Theorem on $\closedint c d$.
Hence:
:$\exists \xi \in \openint c d: \map {f'} \xi = \dfrac {\map f d - \map f c} {d - c}$
Let $f$ be such that:
:$\forall x \in \openint a b: \map {f'} x \ge 0$
Then:
:$\map {f'} \xi \ge 0$
and hence:
:$\map f d \ge \map f c$
Thus $f$ is increasing on $\closedint a b$.
== Sources ==
* : $\S 12.7 \ \text {(i)}$
* 
* 
Category:Increasing Real Functions
Category:Real Functions
Category:Differential Calculus
Real Function with Strictly Negative Derivative is Strictly Decreasing

== Theorem ==
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.
Let:
:$\forall x \in \openint a b: \map {f'} x < 0$
Then $f$ is strictly decreasing on $\closedint a b$.
== Proof ==
Let $c, d \in \closedint a b$ such that $c < d$.
Then $f$ satisfies the conditions of the Mean Value Theorem on $\closedint c d$.
Hence:
:$\exists \xi \in \openint c d: \map {f'} \xi = \dfrac {\map f d - \map f c} {d - c}$
Let $f$ be such that:
:$\forall x \in \openint a b: \map {f'} x < 0$
Then:
:$\map {f'} \xi < 0$
and hence:
:$\map f d < \map f c$
Thus $f$ is strictly decreasing on $\closedint a b$.
== Sources ==
* : $\S 12.7 \ \text {(ii)}$
* 
* 
Category:Strictly Decreasing Real Functions
Category:Real Functions
Category:Differential Calculus
Real Function with Strictly Negative Second Derivative is Strictly Concave

== Theorem ==
Let $f$ be a real function which is twice differentiable on the open interval $\openint a b$ such that: 
:$\map {f} x < 0$ for each $x \in \openint a b$. 
Then $f$ is strictly concave on $\openint a b$ its second derivative $f < 0$ on $\openint a b$.
== Proof ==
From Real Function is Strictly Concave iff Derivative is Strictly Decreasing, $f$ is strictly concave $f'$ is strictly decreasing.
Since $f < 0$, we have that $f'$ is strictly decreasing from Real Function with Strictly Negative Derivative is Strictly Decreasing.
== Also see ==
* Second Derivative of Convex Real Function is Non-Negative
* Real Function with Strictly Positive Second Derivative is Strictly Convex
* Second Derivative of Concave Real Function is Non-Positive
Category:Differential Calculus
Category:Concave Real Functions
Real Function with Strictly Positive Derivative is Strictly Increasing

== Theorem ==
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.
Let:
:$\forall x \in \openint a b: \map {f'} x > 0$
Then $f$ is strictly increasing on $\closedint a b$.
== Proof ==
Let $c, d \in \closedint a b$ such that $c < d$.
Then $f$ satisfies the conditions of the Mean Value Theorem on $\closedint c d$.
Hence:
:$\exists \xi \in \openint c d: \map {f'} \xi = \dfrac {\map f d - \map f c} {d - c}$
Let $f$ be such that:
:$\forall x \in \openint a b: \map {f'} x > 0$
Then:
:$\map {f'} \xi > 0$
and hence:
:$\map f d > \map f c$
Thus $f$ is strictly increasing on $\closedint a b$.
== Sources ==
* : $\S 12.7 \ \text {(i)}$
* 
* 
Category:Strictly Increasing Real Functions
Category:Real Functions
Category:Differential Calculus
Real Function with Strictly Positive Second Derivative is Strictly Convex

== Theorem ==
Let $f$ be a real function which is twice differentiable on the open interval $\openint a b$ such that: 
:$\map {f} x > 0$ for each $x \in \openint a b$.
Then $f$ is strictly convex on $\openint a b$.
== Proof ==
From Real Function is Strictly Convex iff Derivative is Strictly Increasing, $f$ is strictly convex $f'$ is strictly increasing.
Since $f > 0$, we have that $f'$ is strictly increasing from Real Function with Strictly Positive Derivative is Strictly Increasing.
== Also see ==
* Second Derivative of Convex Real Function is Non-Negative
* Second Derivative of Concave Real Function is Non-Positive
* Real Function with Strictly Negative Second Derivative is Strictly Concave
Category:Differential Calculus
Category:Convex Real Functions
Real Inner Product Space is Complex Inner Product Space

== Theorem ==
Let $\struct{V, \innerprod \cdot \cdot}$ be a real inner product space.
Then $\struct{V, \innerprod \cdot \cdot}$ is a complex inner product Space.
== Proof ==
This follows immediately from:
* 
* 
* Real Inner Product is Complex Inner Product
== Also see ==
* Real Semi-Inner Product Space is Complex Semi-Inner Product Space
Category:Inner Product Spaces
Real Inner Product is Complex Inner Product

== Theorem ==
Let $V$ be a vector space over a real subfield $\GF$.
Let $\innerprod \cdot \cdot: V \times V \to \GF$ be a real inner product.
Then $\innerprod \cdot \cdot$ is a complex inner product
== Proof ==
This follows immediately from:
* 
* 
* Complex Number equals Conjugate iff Wholly Real
== Also see ==
* Real Inner Product Space is Complex Inner Product Space
* Real Semi-Inner Product is Complex Semi-Inner Product
Category:Inner Product Spaces
Real Interval is Bounded in Real Numbers

== Theorem ==
Let $\R$ be the real number line considered as an Euclidean space.
Let $a, b \in \R$.
Let $\II$ be one of the following real intervals:
where $b \ge a$.
Then $\II$ is bounded in $\R$.
== Proof ==
Consider the open $\epsilon$-ball $\map {B_\epsilon} a$ where $\epsilon = b + 1 - a$.
As $b \ge a$ we have that $b + 1 > a$ and so $\epsilon > 0$.
Let $x \in \II$.
Then, whatever type of real interval $\II$ actually is, $z \ge a$ and $x \le b$.
As $\epsilon > 0$ it follows that $x > a - \epsilon$.
Also:
That is:
:$a - \epsilon < x < a + \epsilon$
and so:
:$x \in \map {B_\epsilon} a$
The result follows by definition of bounded space.
== Sources ==
* : Chapter $\text{III}$: Metric Spaces: Compactness
Category:Real Intervals
Real Inverse Hyperbolic Sine Function is Bijection

== Theorem ==
The real inverse hyperbolic sine is a bijection.
== Proof ==
From Hyperbolic Sine is Bijection over Reals and by definition of bijection, we have that $\sinh$ admits an inverse function over $\R$.
From:
:Domain of Bijection is Codomain of Inverse
:Codomain of Bijection is Domain of Inverse
the domain and image of hyperbolic sine over $\R$, is $\R$.
Category:Inverse Hyperbolic Sine
Real Linear Functional is Real Part of Unique Linear Functional

== Theorem ==
Let $X$ be a vector space over $\C$. 
Let $g : X \to \R$ be a $\R$-linear functional.
Then there exists a unique $\C$-linear functional $f : X \to \C$ such that: 
:$\map g x = \map \Re {\map f x}$ 
for each $x \in X$. 
Further: 
:$\map f x = \map g x - i \map g {i x}$
== Proof ==
Define $f : X \to \C$ by: 
:$\map f x = \map g x - i \map g {i x}$
for each $x \in X$. 
Then for $\lambda, \mu \in \R$ and $x, y \in X$ we have: 
So $f$ is $\R$-linear. 
To show that $f$ is $\C$-linear, we need to show that:
:$\map f {i x} = i \map f x$ 
for each $x \in X$. 
We have: 
Then for $\alpha + i\beta, \gamma + i\delta \in \C$ and $x, y \in X$ we have: 
So $f : X \to \C$ is a linear functional with:
:$\map g x = \map \Re {\map f x}$
Further, from Linear Functional on Complex Vector Space is Uniquely Determined by Real Part, we have that $f : X \to \C$ is unique.
Category:Linear Functionals
Real Linear Subspace Contains Zero Vector

== Theorem ==
Let $\mathbb W \subseteq \R^n$ such that $\mathbb W$ is a linear subspace of $\R^n$.
Then $\mathbb W$ contains the zero vector:
:$\mathbf 0_{n \times 1} = \begin{bmatrix}
0 \\ 0 \\ \vdots \\ 0
\end{bmatrix} \in \mathbb W$
== Proof ==
This is a consequence of Vector Subspace of Real Vector Space.
== Also see ==
* Zero Subspace is Subspace
Category:Linear Algebra
Real Logarithm is Completely Additive

== Theorem ==
Let $\log_b: \R_{>0} \to \R$ be the real logarithm to base $b$.
Then $\log_b$ is completely additive.
== Proof ==
From Sum of General Logarithms:
:$\log_b x + \log_b y = \log_b \left({x y}\right)$
The result follows from the definition of complete additivity.
Category:Logarithms
Category:Number Theory
Real Multiplication Distributes over Addition

== Theorem ==
The operation of multiplication on the set of real numbers $\R$ is distributive over the operation of addition:
:$\forall x, y, z \in \R:$
::$x \times \paren {y + z} = x \times y + x \times z$
::$\paren {y + z} \times x = y \times x + z \times x$
== Algebraic Proof ==
== Geometric Proof ==
== Sources ==
* : Chapter $1$: The Real and Complex Number Systems: $\text{1-2}$ Arithmetical properties of real numbers: Axiom $3$
* : $\S 3.6$: Reference Formulae
* : Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields
* : Chapter $1$: Real Numbers and Functions of a Real Variable: $1.2$ Operations with Real Numbers: $(5)$
Category:Real Multiplication
Category:Real Addition
Category:Examples of Distributive Operations
Category:Real Multiplication Distributes over Addition
Real Multiplication Distributes over Addition/Algebraic Proof

== Theorem ==
== Proof ==
From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.
Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}, z = \eqclass {\sequence {z_n} } {}$, where $\eqclass {\sequence {x_n} } {}$, $\eqclass {\sequence {y_n} } {}$ and $\eqclass {\sequence {z_n} } {}$ are such equivalence classes.
From the definition of real multiplication, $x \times y$ is defined as:
:$\eqclass {\sequence {x_n} } {} \times \eqclass {\sequence {y_n} } {} = \eqclass {\sequence {x_n \times y_n} } {}$
From the definition of real addition, $x + y$ is defined as:
:$\eqclass {\sequence {x_n} } {} + \eqclass {\sequence {y_n} } {} = \eqclass {\sequence {x_n + y_n} } {}$
Thus:
By Real Addition is Commutative and Real Multiplication is Commutative, it follows that:
:$\paren {y + z} \times x = \paren {y \times x} + \paren {z \times x}$
Category:Real Multiplication Distributes over Addition
Real Multiplication Distributes over Addition/Geometric Proof

== Theorem ==
== Proof ==
:
Let $A$ and $BC$ be two straight lines.
Let $BC$ be cut at random at points $D$ and $E$.
Then the rectangle contained by $A$ and $BC$ is equal to the sum of the rectangles contained by $A$ and $BD$, by $A$ and $DE$, and by $A$ and $EC$, as follows:
Construct $BF$ perpendicular to $BC$.
Construct $BG$ on $BF$ equal to $A$.
Construct $GH$ through $G$ parallel to $BC$.
Construct $DK, EL, CH$ through $D, E, C$ parallel to $BG$.
Then we have that:
: $\Box BCHG = \Box BDKG + \Box DELK + \Box ECHL$
Now $\Box BCHG$ is the rectangle contained by $A$ and $BC$, because it is contained by $BC$ and $BG$, and $BG = A$.
Similarly, from Opposite Sides and Angles of Parallelogram are Equal:
: $\Box BDKG$ is the rectangle contained by $A$ and $BD$, because it is contained by $BD$ and $BG = A$
: $\Box DEKL$ is the rectangle contained by $A$ and $DE$, because it is contained by $DE$ and $DK$, and $DK = A$
: $\Box ECHL$ is the rectangle contained by $A$ and $EC$, because it is contained by $EC$ and $EL$, and $EL = A$.
Hence the result.
== Sources ==
* : Book $\text{II}$. Propositions
Category:Real Multiplication Distributes over Addition