A weak choice principle is introduced that is implied by both countable choice and the law of excluded middle. This principle suffices to prove that metric independence is the same as linear independence in an arbitrary normed space over a locally compact field, and to prove the fundamental theorem of algebra. This paper is written in the context of intuitionistic logic, that is, with no implicit appeal to the law of excluded middle. Following Bishop, we use two seemingly negative expressions in a positive sense. By a nonempty set we mean a set that has a member (an inhabited set), rather than one that cannot be empty; and for elements and of a metric space, the notation x =, y stands for apartness: there exists a positive rational number bounding the distance d(x, y) away from zero. A nonempty subset S of a metric space is located if, for each point in the space, and e > 0, there exists so in S such that d(x, so) 0. In the proof, Bishop tacitly uses countable choice, possibly even dependent choice. Bishop's construction suggests the following definition: Y is strongly reflective if for each there exists Yo in Y such that if =, yo, then is bounded away from Y. Then Bishop's construction shows Bishop's principle: a nonempty, complete, located subset of a metric space is strongly reflective. From Bishop's principle it follows that if k is a locally compact field, then any two norms on k' are equivalent (see [3, Theorem XII.4.2]). Equivalently, metric independence and linear independence are the same in any normed space over k. Using the law of excluded middle, it is easy to show that any nonempty closed subset of a metric space is strongly reflective: let yo = if is in Y, and let yo Received by the editors January 26, 1998 and, in revised form, October 29, 1998. 1991 Mathematics Subject Classification. Primary 03F65, 03E25.