[see also: both]
In either case, it is clear that ...... [= In both cases]
Now equate the coefficients of $x^2$ 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.
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