1. Then x is efficient for p if, and only if, for each i ? N such that p 1i = p 2i =;?;= p ni , x i = p 1i
2. and since y = x, there is i ? N \N * such that y i = x i . Since i ? N \N * , there is j ? N such that p ji = x i . Then this contradicts that y P j x. (Necessity) Let p ? D N and x ? X. Assume that x is efficient for p. Suppose, by contradiction;? D N;Assume that for each i ? N such that p 1i = p 2i