It is already possible to do some of this with prooftrees, though due to a bug it is not possible to start line numbering at 6 with the published version yet.
If you are careful, you can use the split here style included in my other answer and based on Sašo Živanović's code to split a prooftrees prooftree once, interjecting some explanation or a page break or whatever in between.
Note: This WILL break. I don't know where, but there is absolutely no question that it will break. It is barely tested on this proof, let alone others.
Indeed, I discovered 2 existing bugs in prooftrees playing with this, even without splitting the proof into pieces. (These will be fixed shortly after the next version of Forest is pushed to CTAN.)
Caveat emptor...
For example:
\documentclass{article}
\usepackage{prooftrees,amssymb}
% Sašo Živanović: https://tex.stackexchange.com/a/296771/
\def\hiddenparcommand{\par}
\forestset{%
declare count register={split here level},
declare toks register={split here interject},
split here level'=-1,
split here interject={},
to widest/.style={
tikz+={\path (\forestregister{tempdima}, \forestoption{y}) -- (\forestregister{tempdimb}, \forestoption{y});},
},
split here/.style={%
split here level'/.option=level,
split here interject={#1},
split tree
},
split tree/.code={%
\forestset{%
draw tree stage/.style={
for root'={
tempdima/.min={x()+min_x()}{tree},
tempdimb/.max={x()+max_x()}{tree},
for tree={%
to widest,
},
},
for nodewalk/.wrap pgfmath arg={{draw tree processing order/.style={level<=####1}}{}}{split_here_level},
for root'={draw tree},
TeX/.wrap pgfmath arg={\bigskip\hiddenparcommand ####1\hiddenparcommand}{split_here_interject},
for nodewalk/.wrap pgfmath arg={{draw tree processing order/.style={level>=####1}}{}}{(split_here_level)+1},
for root'={draw tree},
},
}
}
}
\begin{document}
\begin{prooftree}
{
line no shift=5,
just refs right,
}
[Fa, name=fa
[{\lnot((Fa \land \forall y (Fy \supset y=a))} \land Ga), checked
[{\lnot(Fa \land \forall y (Fy \supset y=a))}, checked, split here={So far, so automatic. The right hand branch closes, for we have}
[\lnot Fa, just={from:!u}, close={:!uuu,!c}
]
[{\lnot \forall y (Fy \supset y=a)}, checked
[{\exists y \lnot (Fy \supset y=a)}, just={from:!u}, checked=b
[{\lnot (Fb \supset b=a)}, just={from:!u}
]
]
]
]
[\lnot Ga
[Fa \supset Ga, just=from 3
[\lnot Fa, close={:fa,!c}]
[Ga, close={:!uu,!c}]
]
]
]
]
\end{prooftree}
\end{document}

prooftreescould not typeset this. If people actually use it and seem interested, the possibility of splitting trees is something I think would be a useful feature. But I'd rather concentrate on getting the current stuff to work without bugs at the moment. (I'm sure there are bugs. If people use it, they'll no doubt report them.) – cfr Mar 23 '16 at 01:11planis too strong. Haven't really thought about text in between. Splitting at page breaks will not be likely to be automatic. I'd expect to implement a manual option to split a tree at a page break. But, as I say, it may turn out nobody uses even the current version, in which case there won't be any point. And, besides, I don't want to try implementing anything new in it right now. I don't have any confidence in it as it is. There are bound to be cases I've not tried, bugs I've not found. Those get priority. – cfr Mar 23 '16 at 01:15