Original proof; classical forall x: Calgary TFL rules
Premises: (P -> R) and (Q -> R); P or Q. Goal: R.
1 (P -> R) and (Q -> R)  Premise
2 P or Q                  Premise
3 P -> R                  and E 1
4 Q -> R                  and E 1
5 P                       AS, first case
6 R                       ->E 3,5
7 Q                       AS, sibling second case
8 R                       ->E 4,7
9 R                       or E 2,5-6,7-8
Case ranges: 5-6 and 7-8; both endpoints R, both at the same outer level.
Invalid mismatched case inference: (P -> R) and (Q -> S), P or Q therefore R.
Countervaluation: P=F,Q=T,R=F,S=T; premises T, conclusion F.