then under the assumptions for the producers given above (especially the "no arbitrarily large free lunch" assumption), is bounded for any (proof omitted). Thus the first requirement is satisfiable. Define the set of attainable individual production plans to be then under the assumptions for the producers given above (especially the "no arbitrarily large transformations" assumption), is bounded for any (proof omitted). Thus the second requirement is satisfiable. The two requirements together imply that the restriction is not a real restriction when the production plans and consumption plans are "interior" to the restriction. At any price vector , if , then exists and is equal to . In other words, if the production plan of a restricted producer is interior to the artificial restriction, then the unrestricted producer would choose the same production plan. This is proved by exploiting the second requirement on . If all , then the restricted and unrestricted households have the same budget. Now, if we also have , then exists and is equal to . In other words, if the consumption plan of a restricted household is interior to the artificial restriction, then the unrestricted household would choose the same consumption plan. This is proved by exploiting the first requirement on . These two propositions imply that equilibria for the restricted market are equilibria for the unrestricted market: Existence of general equilibrium As the last piece of the construction, we define Walras's law: The unrestricted market satisfies Walras's law at iff all are defined, and , that is, The restricted market satisfies Walras's law at iff . Walras's law can be interpreted on both sides: On the side of the households, it is saying that the aggregate household expenditure is equal to aggregate profit and aggregate income from selling endowments. In other words, every household spends its entire budget. On the side of the producers, it is saying that the aggregate profit plus the aggregate cost equals the aggregate revenue. Note that the above proof does not give an iterative algorithm for finding any equilibrium, as there is no guarantee that the function is a contraction. This is unsurprising, as there is no guarantee (without further assumptions) that any market equilibrium is a stable equilibrium. Uzawa equivalence theorem (Uzawa, 1962) showed that the existence of general equilibrium in an economy characterized by a continuous excess demand function fulfilling Walras’s Law is equivalent to Brouwer fixed-Point theorem.