[see also: both
In either case, it is clear that...... [= In both cases]
Now equate the coefficients of x2 at either end of this chain of equalities.
By Corollary 2, distinct 8-sets have either zero, two or four elements in common.
Each f can be expressed in either of the forms (1) and (2).
The two classes coincide if X is compact. In that case we write C(X) for either of them.
Either f or g must be bounded.
Any map either has a fixed point, or sends some point to its antipode.
But B is not divisible, hence C cannot be divisible either.