This paper uses the Hartman-Stampacchia theorems as primary tool to prove the Gale-Nikaido-Debreu lemma. It also establishes a full equivalence circle among the Hartman Stampacchia theorems, the Gale-Nikaido-Debreu lemmas, and Kakutani and Brouwer fixed point theorems.

The second welfare theorem and the core-equivalence theorem have been proved to be fundamental tools for obtaining equilibrium existence theorems, especially in an infinite dimensional setting. For well-behaved exchange economies that we call proper economies, this paper gives (minimal)...