Let $R$ be a ring and $A$ a right $R$-module. [Not: “Let $R$ a ring”]
Let $f$ satisfy (2). [Not: “Let $f$ satisfies (2)”, nor “Let $f$ verify (2)”.]
Let $f$ be the linear form $g\mapsto (m,g)$.
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 ......