stage ${\bf 1}$ these are ordinary transformations without a “hidden” multidimensional parameter $u$, while at the stage $R$ these are smooth transformations with a “hidden” parameter.

The relation of causal ordering on $R^4$ can be defined with the help of the formula $$\forall x\in R^4\exists P_x\subset R^4(x\in P_x \ \& \
(\forall y\in P_x
\Rightarrow P_y\subset P_x)\  \&
\ (\forall u\neq v\Rightarrow P_u\neq P_v)),$$ where $F\subset G$ means, that a type $F$ is a subtype of type $G$ and one is interpreted in topos ${\bf Set^{L^{op}}}$ as subfunctor [@MR]. Essentially, at stage $\ell A$ a set $P_x(\ell A)$ consist of classes $C^{\infty}$-smooth vector-functions $$((p_x^0(u),p_x^1(u),p_x^2(u), p_x^3(u)):\R^n\rightarrow\R^4)mod \ I;$$ moreover a term $x$ is interpreted as an element $P_x(\ell A)$, i.e. class $(X^0(u),X^1(u),$ $X^2(u),$ $X^3(u))mod \ I$. Since $R^4$ at every stage can equip the vector structure, i.e. convert in vector space, then naturally to think, that set $P_0(\ell A)$ modulo $I$ is semigroup with respect to addition, where $0$ at stage $\ell A$ is class $mod \ I$.

The causal automorphisms $f:R^4\rightarrow R^4$ at stage $\ell A$ are defined by relations of the form $$F(u, X(u)mod \ I + P_0(\ell A) )mod \ \pi^*(I)
= F(u, X(u)mod \ I)mod \ \pi^*(I) 