Equivalence of Definitions of Strictly Progressing Mappings

Theorem
Let $g$ be a mapping.

$(1)$ implies $(2)$
Let $g$ be a strictly progressing mapping by definition $1$.

Then by definition:
 * $\forall x \in \Dom g: x \subsetneqq \map g x$

Hence :
 * $\forall x \in \Dom g: x \subseteq \map g x$

and so $g$ is a progressing mapping.

Also :
 * $\forall x \in \Dom g: x \ne \map g x$

and so $g$ has no fixed points.

Thus $g$ is a strictly progressing mapping by definition $2$.

$(2)$ implies $(1)$
Let $g$ be a strictly progressing mapping by definition $2$.

Then by definition:
 * $\forall x \in \Dom g: x \subseteq \map g x$

and also:
 * $\forall x \in \Dom g: x \ne \map g x$

That is:
 * $\forall x \in \Dom g: x \subsetneqq \map g x$

Thus $g$ is a strictly progressing mapping by definition $1$.