Section Req_bool.
Definition Req_bool x y :=
match Rcompare x y with
| Eq => true
| _ => false
end.
Inductive Req_bool_prop (x y : R) : bool -> Prop :=
| Req_bool_true_ : (x = y)%R -> Req_bool_prop x y true
| Req_bool_false_ : (x <> y)%R -> Req_bool_prop x y false.

Theorem Req_bool_false :
forall x y,
(x <> y)%R -> Req_bool x y = false.