The train company has absolute evidence of one offence, and can prosecute on that. (They probably have evidence of many more, given the surveillance, but let's ignore that for now.)
The way it works is that they will say "Either we prosecute you for the one offence, or you accept our settlement offer, which involves paying for all the instances where we are pretty sure that you have evaded the correct fare. If on the day the person was stopped they bought a ticket from (say) Brighton to Preston Park as part of doughnutting, and another from London Bridge to Blackfriars, and the investigators see another 50 instances where those two fares were purchased on the same day than they can be pretty confident that it is reasonable to base their settlement offer on the traveller having evaded the correct fare on those 50 occasions too.
The evader then has the option of going to court for the one offence (which most people want to avoid for obvious reasons), or paying what is asked of them to settle out of court.
As I said, in this instance it's much more likely that GTR could build a case that would hold in court that the evasion was undertaken on more than just the one day they were stopped.