C. Then and by IH (B (B C); this finally yields the other cases to the reader. C) which implies C). 3. 2A. 4 Do the remaining cases. 3. DEFINITION.
The set of assumptions of the same form with the same marker forms an assumption class. Distinct formulas must have distinct markers. We permit empty assumption classes! Assumptions may be closed; assumption classes are always closed "en bloc", that is to say, at each inference, either all assumptions in a class are closed, or they are all left open. Closure is indicated by repeating the marker(s) of the class(es) at the inference. For ease in the exposition, we shall reserve u, y, w for assumption markers, and x, y, z for individual variables.
Basic Proof Theory by A. S. Troelstra, H. Schwichtenberg