A double angle formula: $sin 2x = 2 sin x cos x$.

A $5 by 8$ chessboard contains $5 times 8 = 40$ little squares.

We have $abs u+v <= abs u + abs v$ for all $u,v in CC$,
while $norm(vec u + vec v)^2 = norm(vec u)^2 + norm(vec v)^2$ only if $vec u perp vec v$.

If $a x^2 + b x + c = 0$ then $x = (-b pm sqrt(b^2 - 4 a c))/2a$,
and
$$"if" a x^2 + b x + c = 0 "then" x = (-b pm sqrt(b^2 - 4 a c))/2a
$$

$$zeta(s) = \sum_n=1^infty 1/n^s = prod_p (1-p^-s)^-1$$


$$
abs x = cases:
    x if x > 0
    0 if x = 0
    -x otherwise
$$
Another line.

$$
derivation:
int_A^T 1/(x log x llog x) dx = lllog T + bigO(1)

asymp lllog T
$$
as $T to infty$.

$$
system:
   a + b = c < 8
      +55

   a^2 <= 77 b^3 + 12

   x^n + y^n = z^n
$$

+++++++++++++++=

Volker's MathML generator
https://speech-rule-engine.github.io/semantic-tree-visualiser/visualise.html

$tan x = (sin x)/(cos x)$

A double angle formula: $sin 2x = 2 sin x cos x$.

A $5 by 8$ chessboard contains $5 times 8 = 40$ little squares.

We have $abs u+v <= abs u + abs v$ for all $u,v in CC$,
while $norm(vec u + vec v)^2 = norm(vec u)^2 + norm(vec v)^2$ only if $vec u perp vec v$.

The Chebyshev polynomials of the first kind, $T_n$, are defined by
$T_n(cos theta) = cos(n theta)$.  They are orthogonal with respect
to the inner product
$$
<f,g> = \int_-1^1 f(x) g(x) dx/sqrt[1-x^2]
$$
[[above weight in the integral not parsing correctly: consider wrapping the sqrt)

If $y'' = c^2 y$ then $y(x) = A e^(c x) +b e^{-c x}$ for some $A,b in CC$

$(f g)' = f' g + f g'$

$f(g(x))' = f'(g(x)) g'(x)$

If $A$ is a permutation matrix then $det A = +-1$ 

If $g \in G$ then $order g | order G$

If $a$ and $b$ are finite sets, then $card(a product b) = card a times card b$

If $a x^2 + b x + c = 0$ then $x = (-b pm sqrt(b^2 - 4 a c))/2a$,
and
$$"if" a x^2 + b x + c = 0 "then" x = (-b pm sqrt(b^2 - 4 a c))/2a
$$

$$
derivation:
int_A^T 1/(x log x llog x) dx = lllog T + bigO(1)

asymp lllog T
$$
as $T to infty$.

${ f:RR to ZZ | f(0) <= 5}$

$int_0^infty x/(e^(sqrt x)) dx$

$sigma(n) = sum_d|n d$, and more generally, $sigma_k(n) = sum_d|n d^k$

$J_0^2(x) + a_i^45$

$f(1/2) + f(1//2) = e^pi$

$lim_[x to infty] (sin x)/x = 0$

${3,4} subset ZZ$, and $(3,4) in RR^2$, and $(3, 4) subset RR$.

$80 cent + 20 cent = dollar1$

$emptyset = {x in ZZ | 0 < x < 1}$

$sum_n=1^N n = N(N+1)/2$

$sin^2 x + cos^2 x = 1$

The Fibonacci sequence is defined by the recursion
$F_(n+1) = F_n + F_(n-1)$, with initial conditions $F_0 = F_1 = 1$.
In closed form:
$$
F_n = floor(1/sqrt5 \phi^n) "where" phi = (1 + sqrt(5))/2  "is the golden ratio".
$$

We use the following notation for integration along a vertical
line in the complex plane:
$$
  int_((c)) f(s) ds := int_(c-i infty)^(c + i infty) f(s) ds .
$$
For example, we have the Mellin inversion pair:
$$
Gamma(s) = \int_0^infty e^-x x^s dx/x
"and"
e^z = 1/(2 pi i) int_((1)) Gamma(s) z^-s ds .
$$

$−(2n+2) < −a_n < -2n$


$$(k+1)(k/2 + 1)$$

If $a | b$ then $c a | c b$.

$e^{i theta} = cos(theta) + i sin(theta)$

$ZZ subset RR intersect QQ$

$y^2 = x^3 + a x + b "with" a, b in QQ$

$18 cong 7 mod 11$

$x^2 cong 0 "or" 1 mod 4$

$$ p | prod_n=1^K a_n "implies" p | a_n "for some" n $$

$(p-1)! cong -1 mod p$

If $a !cong 0 mod b$ then $a^(p-1) congruent -1 mod p$

$$zeta(s) = \sum_n=1^infty 1/n^s = prod_p (1-p^-s)^-1$$

$1 != 2$

--------------

$$
abs x = cases:
    x if x > 0
    0 if x = 0
    -x otherwise
$$
Another line.

$$
system:
   a + b = c

   a^2 < 77 b^3 + 12
$$

$$
system:
   a + b = c < 8
      +55

   a^2 <= 77 b^3 + 12

   x^n + y^n = z^n
$$
Some words.


Markup for  K/Q ?



Gegenbauer polynomial: $G_n^(alpha)(x)$

Chebyshev polynomial: $P_n(x)$

*** need the above to be recocnised as functions
??? maybe funcitonwrap ?
??? also rething opwrap:  should it have the argument in it?
         (answer: no, because it is hard to parse what is the argument)

$$zeta(s) = \sum_n=1^infty 1/n^s = prod_p (1-p^-s)^-1$$

$sum_(n=1)^oo x^n = frac1{1-x}$$

$12x^2-5x^2+19x-100/99-p^y$

 need a test to compare \frac{-x^2}{1}  and -\frac{x^2}{2} 

$Gamma(s) = \int_0^infty e^-x x^s dx/x$


$e^x^2/2 + e^-x^2 / 2 + e^[x^2]/2 + e^[x^2] / 2  + e^(-x^2 / 2)  e^(- x^2 / 2) $

$sin x^2$ or $sin (1+x)^2$

$x^{(} + y^{)} = z^[$ 

Suppose $x > 0$, and $y < 0$.