English
There is an AddAction of the real numbers on the upper half-plane given by translation in the real direction: x +ᵥ z = mk(x + z) with Im(z) > 0; this action satisfies identity and associativity rules.
Русский
Существует действие сложения вещественных чисел на верхней полуплоскости: x +ᵥ z = mk(x + z) при Im(z) > 0; это действие удовлетворяет тождеству и ассоциативности.
LaTeX
$$There is an AddAction Real UpperHalfPlane: vadd x z = mk (x + z) with z.im > 0; vadd is identity and associative.$$
Lean4
instance : AddAction ℝ ℍ where
vadd x z := mk (x + z) <| by simpa using z.im_pos
zero_vadd _ := Subtype.ext <| by simp [HVAdd.hVAdd]
add_vadd x y z := Subtype.ext <| by simp [HVAdd.hVAdd, add_assoc]