Referring to G. Perelman, Proof of the soul conjecture by Cheeger and Gromoll. Given a distance-nonincreasing retraction $P$ from an open complete manifold of nonnegative curvature onto its soul $S$, one wants to prove

(A) $P(\operatorname{exp}_xtν)=x$ for every $x∈S$ and $t≥0$.

(B) For any geodesic $\gamma \subset S$ and any unit vector field $\nu$ in its normal bundle which is parallel along $\gamma$, the "horizontal" curves $\gamma_t, \gamma_t(u)=\exp _{\gamma(u)}(t \nu)$, are geodesics, filling a flat totally geodesic strip $(t \geq 0)$. Moreover, if $\gamma\left[u_0, u_1\right]$ is minimizing, then all $\gamma_t\left[u_0, u_1\right]$ are also minimizing.

Suppose this is true up to $t=l$ , and consider the function $$ f(r)=\max \left\{\left|x P\left(\exp _x(l+r) \nu\right)\right| \mid x \in S, \nu \in S N_x(S)\right\} $$ (1) He said that it's clear that $f$ is Lipschitz continuous, but I can't get understand it;

(2) The outline of the proof is "it is sufficient to check that if (A) and (B) hold for $0 \leq t \leq l$ for some $l \geq 0$, then they continue to hold for $0 \leq t \leq l+\varepsilon(l)$, for some $\varepsilon(l)>0$. ". However, can the process leads us to this conclusion is true for all $l\in (0,+\infty)$?

(3) By the way, the existence of such a distance nonincreasing retraction of $M$ onto $S$ is due to Sharafutdinov. Can anyone provide any references in English? Thanks in advance.