This a ratio problem, how much of the season has been used by learning the spell with the focus... and then applying the remainder against the non focused lab total.
[1-(Amount of lab total used to learn the first spell using the focus/lab total with the focus)] * LT without Focus
1-(Amount of lab total used to learn the first spell using the focus/lab total with the focus) is the amount of time left in the season.
Art scores of Te=10, Fo=10, +3 Int +3 MT +3 aura.
LT w/ Focus=10+10+10+3+3+3=39
LT w/o Focus=10+10+3+3+3=29
Using the focus, they invent a 10th level spell, without a lab text, needing 20 points. 1-20/39=.487, which is the amount of time left remaining in the season, essentially.
So .487*29=14.1, so learn a 10th level spell with a lab text or a 5th level spell without a lab text.
Although I agree that this is the 'math-wise correct approach' approach, I believe I would rule as Xavi proposed in the above. If not for anything else, then to simplify. If the bonus is really important though or you would like to do things accurately I tottally agree with Jonathan. That said I'm not sure 10+10+3+3+3 equals 27, but that's beside the point
Since they're all addition, it doesn't matter when the Focus gets applied, which means if it all fits, then it all fits. However, a non-focus spell has to fit in the lower LT as well.
Yes. If Spell 1 > Basic LT, then no.
Again, yes. In fact, I'd go a step farther -- If the player tried this IMS, then forever after, if his magus has LT-with-Focus leftover, I would require him to use it on a spell which includes his Focus. My reasoning: a Focus is not just a mathematical modifier, it is a statement about the wizard's personality and interests. They're always thinking about how things apply to their focus. They're constantly doodling notes in the side margins of their Lab Texts about the implications to their focus.
They are, you might say, focused on the subject.
And besides, the gorram players deserves to do the extra math if he's going to bug me with this sort of question. Call it a grudge
Yeah, because hermetics are so underpowered that they need every help that the alpha SG can give them to be competitive
But yes, you can not like it easily. We ended up opting for hard and fast rules in the end, even if they are harsher, basically because we dislike number crunching, so ease of use is superior to detailed multipliers and complicated optimal calculations