I am reading Yde Venema's text on Temporal Logic found here: https://staff.fnwi.uva.nl/y.venema/papers/TempLog.pdf
Temporal Logic differs from regular Modal Logic by having two non-classical connectives, namely 'henceforth' G and 'hitherto' H. In this question, only G matters. The truth value of $G\varphi$ at point $t$ is defined as follows:
$\mathcal{M}, t \models G\varphi \text{ if } \mathcal{M}, s \models \varphi \text{ for all }s\text{ with } t < s$
In section 3.2 Validity and Definability, he introduces how a class of frames $\mathsf{C}$ in $\mathsf{K}$ can be characterised by a formula $\varphi$ by saying:
formula $\varphi$ defines a class of frames $\mathsf{C}$ if for every frame $\mathcal{T}$ in $\mathsf{K}$, $\mathcal{T} \models \varphi$ iff $\mathcal{T}$ belongs to $\mathsf{C}$.
As an example, he works out why the formula $PFq \rightarrow Pq\vee q \vee Fq$ models the calss of frames that have a non-branching future for all time ponits $t\in\mathcal{T}$. Later, there is an off hand remark that, just like in regular modal logic, $Gp \rightarrow GGp$, models transitivity, which I have difficulty proving.
That transitivity implies $\mathcal{T} \models Gp \rightarrow GGp$ is quite trivial, but the other way around, I only mange to rewrite the statement $\mathcal{T} \models Gp \rightarrow GGp$ to
$\left(\forall_{s>t}\; \mathcal{M},s \models p\right) \rightarrow \forall_{s>t}\forall_{j>s}\;\mathcal{M},j \models p$
And I do not see how this implies that $\mathcal{T}$ is transitive.
