Rabin has investigated the difficulty of proving that a set of linear forms is simultaneously positive by the evaluation of analytic functions. In this paper we study this same question under the restriction that each analytic function itself be linear. A complete result is given in the case that the original set of linear forms are simultaneously positive on a subspace having at least one extreme point. Applications are then given. In particular, it is shown that the proof that a real number x 1 is maximal out ol the set {xl .... , x~} requires evaluation of m --I linear forms even if x x is known in advance to be exceeded by at most one xi for 2 ~ i ~_ m. I. INTRODUCTION Let Lx ~ b be a system of linear inequalities over the real field R defined by an m โข d matrix L and x and b column vectors of d components. Let C C R a be a convex set. Then Rabin [1] asks the question: Given x E C, is there a way to prove thatLx ~ b which involves the evaluation of less than m (not necessarily linear) inequalities ? In order to answer this question Rabin defines an independence condition of the system relative to the set C and proves that if m ~ d and this condition is satisfied then if {Pi~}i~=l ~1 is any finite collection of analytic functions each with domain R a and range R such that k (Vx~C)Lx ~ b iff 3i~ Apij(x) >/o; ~=1 then k ~ m. This says, in effect, that, given x ~ C, evaluation of the m linear forms in the original system is the easiest way to determine if Lx ~ b or not. Conversely, he