I am reading Lattice Theory: Foundation written by George Gratzer. In theorem 128,
he said:
Let L be a distributive lattice generated by I. The lattice L is distributive freely generated by I iff the validity in L of a relation of the form $$\bigvee I_0\le \bigvee I_1$$ implies that $I_0 \cap I_1 \neq \emptyset$ for finite non empty subsets $I_0$ and $I_1$ of I.
He proof the "only if" part by using substitutions in $C_2$, but I cannot understand. I know distributive isomorphic to a ring of set so isomorphic to ${C_2}^{|I|}$, and I can proof it by this. But how can just use substitutions in $C_2$? I want to understand this method.