Proof of excess community demand function in Sonnenschein (1973)

Consider the following passage in Sonnenschein (1973; full citation below):

Perhaps because I do not know much about the uniform Lipschitz condition, I do not follow his exposition after the first two sentences of the proof. That is, do the definitions of $g_1^1$ and $g_1^2$ come from some standard result in mathematics? Any intuition here would be appreciated.

Furthermore, what is the source of $(g_1^1, I-pg_1^1)$ and $(g_2^1, I-pg_2^1)$?

Thank you.

Citation:

Sonnenschein, H. 1973. "Do Walras’ identity and Continuity Characterize
the Class of Community Excess Demand Functions?" Journal of Economic Theory 6: 345-354.