I was especially thrown out by the proof on The Transversality Theorem on the point "we want to exhibit a vector $v \in T_x(X)$ such that $df_s(v) - a \in T_z(Z).$"
I understand so far that in order to show that $f_s \pitchfork Z$ we need to show $df_sT_x(v) + T_z(Z) = T_z(Y)$. But why we want to show "a vector $v \in T_x(X)$ such that $df_s(v) - a \in T_z(Z).$"? I guess I lost some general idea here.
I am looking forward a detailed answer which fills up what author omitted.. Thank you!