The following picture contains a theorem. The theorem states some equivalent formula in deterministic frames ( Let M=(W, R, V) be a model. For all w, $v_1$, $v_2$ in W: if Rw$v_1$ and Rw$v_2$, then $v_1=v_2$).
Would you please show why item 6 is provable?
The argument in the quoted text can be outlined as
$$P_{1}\iff P_{2}\iff P_{3}$$
using the conclusion as a hypothesis in the derivation of another one in turn. The same method can be employed to show that
$$P_{4}\iff P_{5}\iff P_{6}$$
In order to show each formula by itself, the frame is required to be not only deterministic, but also reflexive; hence we have to adjoin axiom of $\mathbf{T}$
$$\Box\phi\rightarrow\phi$$
to the base system $\mathbf{K}$. So, the minimal system we shall work is to be a deterministic $\mathbf{T}$; we may call it $\mathbf{DT}$.
The neatest way for such basic formulas is the semantic one. We may reason as follows:
In case that $\Diamond\phi$ is true at $w_{0}$, $\phi$ is to be true at some $w_{1}$.
In case that $\Box\phi$ is true at $w_{0}$, $\phi$ is to be true at $w_{1}$, and if they exist, all the other accessible worlds, $w_{2}, w_{3},\ldots$ which are, by determinism, all identical to $w_{1}$. But this is nothing different than the truth-condition of $\Diamond\phi$. Hence, $\Diamond\phi\rightarrow\Box\phi$.
By transitivity of material implication and the reflexivity (factivity) axiom, we get $\Diamond\phi\rightarrow\phi$.
Several remarks:
(1) By the base system $\mathbf{K}$, we have already $\Box\phi\rightarrow\Diamond\phi$. Thus, actually in $\mathbf{DT}$, the stronger statement $\Box\phi\leftrightarrow\Diamond\phi$ is valid.
(2) $\Box\phi$ is vacuously true at $w_{0}$ if there is no distinct accessible world $w_{1}$.
(3) Notice that we have obtained the formula $\Diamond\phi\rightarrow\Box\phi$ cited as (3) on the way. It is appealing to our intuitions about a deterministic world, stating that if a state of affairs is possible, then it is necessarily the case, and is sometimes taken as an axiom to derive $\Diamond\phi\rightarrow\phi$ and others.