(* What it is to be a maximum *)
 
Definition isMax (r : R) (max : float) :=
  Fbounded b max /\
  (r <= max)%R /\
  (forall f : float, Fbounded b f -> (r <= f)%R -> (max <= f)%R).

Theorem MinExList : forall (r : R) (L : list float), (forall f : float, In f L -> (r < f)%R) \/ (exists min : float, In min L /\ (min <= r)%R /\ (forall f : float, In f L -> (f <= r)%R -> (f <= min)%R)).