Vrftjkry

I have some questions about the arguments used in following excerpt in J. Milnor's " Topology from the differentiable view point":

We have following setting: $f: H^m to N^n$ is a smooth map from a $m$-manifold with boundary $partial H^m$ to a $n$-manifold without boundary. Set $y in N_n $ regular for $f$ and the restriction $f vert _{partial N}$.
This means that $dim T_x f= dim_x (f vert _{partial H^m}) =n$ for every $x in f^{-1}(y)$.

Since the problem is local we assume $N^n= mathbb{R}^n$.

My questions are:

Let $x in pi^{-1}(0) subset f^{-1}(y)$.

1.: Why the tangent space $T_x pi$ equals to the null space (= kernel) of the tangential map $dg_x = df: mathbb{R}^m to mathbb{R}^n$?
(remark: $dg_x = T_xg: T_xH^m to T_{g(x)}mathbb{R}^n$ where $T_xH^m= mathbb{R}^m$ since $H^m$ is a $m$-manifold and $T_{g(x)}mathbb{R}^n = mathbb{R}^n$

2. Why the fact that $f vert _{partial H^m}$ and $f$ are regular implies that $ker(T_x g) not subset mathbb{R}^{m-1} times {0}$?

We know that since $f$ and $f vert _{partial H^m}$ regular, then $ker(T_x f)= m-n$ and $ker(T_x (f vert _{partial H^m}))= m-1-n$ by linear algbra but I don't see how it should imply $pi(ker(T_x f)) neq 0$.

This should answer question 1. I am not sure, immediately, how to answer your second question. I'll make an edit if I think of something.