$S$ - numerical semigroup, generated by $a$, $b$.
$S=(a,b)={s0 < s1 < s2 < .....}$
$c$ - conductor of numerical semigroup: $c=(a-1)(b-1)$.
All elements $x_i >= c$ are incremented by one.
How can I prove it?
My teacher wrote in (see pictures). Which famous formula did he use?
Tell me, what is the formula. Next I will try to prove the theorem itself.
I think you're referring to the ua+vb=1 part? This is an idea I've often heard referred to as "the Euclidean algorithm". Technically this is an algorithm for finding appropriate coefficients, but it's correctness of course also implies the theorem that such coefficients are guaranteed to exist (whenever a and b are coprime).