M.S.

A *straight line* is defined as a function $f: \mathbb{R} \to \mathbb{R}$ (more general fields are available) given by $x \mapsto a x+b$ for some real constants $a, b$. A *point* is defined as a 2-tuple of reals.

to be the definition of the gaps in the current math?

(CM.) - natural numbers are given as an axiom, there is no natural gaps numbers (there is this form, but do not call numbers $(\{0,0\}\cup\{a,a\} a\in N)$

Theorem - natural numbers and natural numbers gaps can be connected in the direction AB (0.1)


PROOF - Number 1 and number $\underline {1}$receives the combined number of $1\underline {1}$ or dup (duž , praznina )



-Number $\underline {1}$ and number 1 receives the combined number of $\underline {1}1$ or dup



-Number 1 and number $\underline {2}$ receives the combined number of $1\underline {2}$ or dup

...
- A basic set of combined natural numbers $K^o=(a_n,\underline{b}_n,a_n\in{N^o},\underline{b}_n\in{N_p^o},(a_n,\underline{b}_n)>0)$


$a_1\underline{b}_1$
$\underline{b}_1a_1$
$a_1\underline{b}_1a_2$
$\underline{b}_1a_1\underline{b}_2$

...

(CM.) - Dup do not know, not know the combined numbers (there is this form, but not numbers $\{0,a\}\cup\{c,c\},\{0,0\}\cup\{a,b\},\{0,b\}\cup\{c,d\},\{0,0\}\cup\{a,b\}\cup\{c,c\},...$ )

Smaug123 - to current mathematics defined dup ?

I don't know what you mean by "dup", I'm afraid - do you mean the ordered pair $(a, a)$, for some $a$? If so, it can be implemented in set notation as $\{a, \{a\} \}$.

dup=straights lines + gaps

I don't see how a dup differs from the tuple (point1, point2, line between point1 and point2)?

Theorem - The contact number is sorted horizontally only be a natural straight line that gives a natural straight line


proof -$1\rightarrow 1$

4${+_1^{\underline0}}$2=2


4${+_1^{\underline1}}$2=(1,1)


4${+_1^{\underline2}}$2=2


4${+_1^{\underline3}}$2=(3,1)


4${+_1^{\underline4}}$2=6 or 4+2=6


$+_1$ - addition rule 1

.

Theorem - The contact numbers is sorted horizontally to be the only one natural straight line that gives a natural straight line , when there are two (more) results between them becomes a gap.


PROOF -$1\rightarrow1(\underline{1})$

$4{+_2^{\underline0}}2=2$
$4{+_2^{\underline1}}2=1 \underline{2}1$
$4{+_2^{\underline2}}2=2$
$4{+_2^{\underline3}}2=3 \underline{1}1$
$4{+_2^{\underline4}}2=6$


+₂ - addition rule 2


(CM.) - No "addition rule 2"
more complex
{0,4}$\cup${6,9}$\cup${11,13} ? {0,3}$\cup${5,10} = {0,1}$\cup${2,3}$\cup${6,7}$\cup${8,10} my solution and notation $4\underline23\underline22{+_{2}^{\underline0}}3 \underline25$=111311​2
{0,4}$\cup${6,9}$\cup${11,13} ? {0,3}$\cup${5,10} = {0,1}$\cup${9,13} my solution and notation $4\underline23\underline22{+_{2}^{\underline1}}3 \underline25$=18​4
{0,4}$\cup${6,9}$\cup${11,13} ? {0,3}$\cup${5,10} ={0,2}$\cup${4,5}$\cup${6,7}$\cup${9,11}$\cup${12,13} my solution and notation $4\underline23\underline22{+_{2}^{\underline2}}3 \underline25$=22111221​1
...

{0,4}$\cup${6,9}$\cup${11,13} ? {0,3}$\cup${5,10} ={0,4}$\cup${6,9}$\cup${11,16} $\cup${18,23} my solution and notation $4\underline23\underline22{+_{2}^{\underline{13}}}3 \underline25$=423252​5