Testing Latex Rendering

Just some random text


$$1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots = \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}, \quad\quad \text{for} |q|<1.$$


$$\begin{aligned} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}$$


embedded latex

More MathJax tests. Embedding formulae inside sentences

The formula $\prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}, \quad\quad \text{for} |q|<1.$ embedded in the middle of a sentence

This expression $\sqrt{3x-1}+(1+x)^2$ is an example of an inline equation.



The question is Prove Universal Instantiation i.e

Prove $(\forall x |: P) \Rightarrow P [ x := E ]$

LHS $\hspace{2 cm } (\forall x |: P)$

= $\hspace{0.5 cm }$ notation expansion

$\hspace{2 cm } (\forall x | true : P)$

= $\hspace{0.5 cm }$ true $\equiv a \vee \neg a$

$\hspace{2 cm } (\forall x | (x = E) \vee (x \neq E) : P)$

= $\hspace{0.5 cm }$ Axiom: Range Split $(\star | R \vee S: P) \equiv (\star | R : P) \star (\star | S : P)$ here $\star = \wedge.$ notated as $\forall$


$\hspace{2 cm } (\forall x | (x = E) : P) \wedge (\forall x | (x \neq E) : P)$

$\Rightarrow \hspace{0.5 cm } Theorem: a \wedge b \Rightarrow a$

$\hspace{2 cm } (\forall x | (x = E) : P)$

= $\hspace{0.5 cm }$ Single Point Axiom $(\star y | (y = F) : Q) \equiv$ Q [ y := F ]

$\hspace{2 cm }$ P [ x := E ] = RHS.
QED