Let R be a ring and A a right R-module.
Let f satisfy (2). [Not: “Let f satisfies (2)”, nor “Let f verify (2)''.]
Let f be the linear form g→ (m,g).
We let T denote the set of......
One cannot in general let A be an arbitrary substructure here.
Letting m tend to zero identifies this limit as H.
As we let t vary, f(t) describes a curve in M.
The desired conclusion follows after one divides by t and lets t tend to 0.
Now, just the fact that F is a homeomorphism lets us prove that......