we don't need to prove anything mathematically, I have enough of that in my engineering courses! The fact is that everytime the inspector drawn a ticket (even for just a supplement) adds a fee of £.10.000! I can tell you for sure, because I had it personally! Once my train (express) was too late and we took a different one (InterCity Night) thinking it was the same, and we had to pay for it (the good thing was that the inspector, at our request, made only 1 ticket for the whole group so we could share the fee).

BTW, the best way is always to check out and ask before leaving.

Ciao Flavio in Rome

The surcharge is not due if you 1) notify the conductor upon boarding; 2) You have a ticket valid for a lower category of trains 3) You are traveling on a non regional trains (regional trains are not only Regionals but also D and iR trains)

So the suggestion in case of doupt on which train to get, is buying a 2nd class ticket for a iR train, then ask the conductor to integrate the fare. Note that if the integration is from 2nd to 1st class condition 3 is not required.